24th Annual European Symposium on Algorithms (ESA 2016), ESA 2016, August 22-24, 2016, Aarhus, Denmark
ESA 2016
August 22-24, 2016
Aarhus, Denmark
European Symposium on Algorithms
ESA
http://esa-symposium.org/
https://dblp.org/db/conf/esa
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Piotr
Sankowski
Piotr Sankowski
Christos
Zaroliagis
Christos Zaroliagis
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
57
2016
978-3-95977-015-6
https://www.dagstuhl.de/dagpub/978-3-95977-015-6
Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers
Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers
Front Matter
Table of Contents
Preface
Programm Commitee
External Reviewers
0:i-0:xxiv
Front Matter
Piotr
Sankowski
Piotr Sankowski
Christos
Zaroliagis
Christos Zaroliagis
10.4230/LIPIcs.ESA.2016.0
Creative Commons Attribution 3.0 Unported license
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2-Connectivity in Directed Graphs (Invited Talk)
We survey some recent results on 2-edge and 2-vertex connectivity problems in directed graphs. Despite being complete analogs of the corresponding notions on undirected graphs, in digraphs 2-vertex and 2-edge connectivity have a much richer and more complicated structure. It is thus not surprising that 2-connectivity problems on directed graphs appear to be more difficult than on undirected graphs. For undirected graphs it has been known for over 40 years how to compute all bridges, articulation points, 2-edge- and 2-vertex-connected components in linear time, by simply using depth-first search. In the case of digraphs, however, the very same problems have been much more challenging and required the development of new tools and techniques.
2-edge and 2-vertex connectivity on directed graphs
graph algorithms
dominator trees
1:1-1:14
Invited Talk
Loukas
Georgiadis
Loukas Georgiadis
Giuseppe F.
Italiano
Giuseppe F. Italiano
Nikos
Parotsidis
Nikos Parotsidis
10.4230/LIPIcs.ESA.2016.1
S. Alstrup, D. Harel, P. W. Lauridsen, and M. Thorup. Dominators in linear time. SIAM Journal on Computing, 28(6):2117-32, 1999.
J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications (Springer Monographs in Mathematics). Springer, 1st ed. 2001. 3rd printing edition, 2002.
A. A. Benczúr. Counterexamples for directed and node capacitated cut-trees. SIAM J. Comput., 24:505-510, 1995.
A. L. Buchsbaum, L. Georgiadis, H. Kaplan, A. Rogers, R. E. Tarjan, and J. R. Westbrook. Linear-time algorithms for dominators and other path-evaluation problems. SIAM Journal on Computing, 38(4):1533-1573, 2008.
K. Chatterjee and M. Henzinger. Efficient and dynamic algorithms for alternating büchi games and maximal end-component decomposition. J. ACM, 61(3):15:1-15:40, 2014. URL: http://dx.doi.org/10.1145/2597631.
http://dx.doi.org/10.1145/2597631
J. Cheriyan and R. Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput., 30(2):528-560, 2000.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms, Second Edition. The MIT Press, 2001.
W. Di Luigi, L. Georgiadis, G. F. Italiano, L. Laura, and N. Parotsidis. 2-connectivity in directed graphs: An experimental study. In Proc. 17th Workshop on Algorithm Engineering and Experiments, pages 173-187, 2015. URL: http://dx.doi.org/10.1137/1.9781611973754.15.
http://dx.doi.org/10.1137/1.9781611973754.15
Ya. M. Erusalimskii and G. G. Svetlov. Bijoin points, bibridges, and biblocks of directed graphs. Cybernetics, 16(1):41-44, 1980. URL: http://dx.doi.org/10.1007/BF01099359.
http://dx.doi.org/10.1007/BF01099359
D. Firmani, G. F. Italiano, L. Laura, A. Orlandi, and F. Santaroni. Computing strong articulation points and strong bridges in large scale graphs. In Proc. 10th Int'l. Symp. on Experimental Algorithms, pages 195-207, 2012.
W. Fraczak, L. Georgiadis, A. Miller, and R. E. Tarjan. Finding dominators via disjoint set union. Journal of Discrete Algorithms, 23:2-20, 2013. URL: http://dx.doi.org/10.1016/j.jda.2013.10.003.
http://dx.doi.org/10.1016/j.jda.2013.10.003
H. N. Gabow. The minset-poset approach to representations of graph connectivity. ACM Transactions on Algorithms, 12(2):24:1-24:73, February 2016. URL: http://dx.doi.org/10.1145/2764909.
http://dx.doi.org/10.1145/2764909
M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., New York, NY, USA, 1979.
L. Georgiadis. Approximating the smallest 2-vertex connected spanning subgraph of a directed graph. In Proc. 19th European Symposium on Algorithms, pages 13-24, 2011.
L. Georgiadis, G. F. Italiano, A. Karanasiou, C. Papadopoulos, and N. Parotsidis. Sparse subgraphs for 2-connectivity in directed graphs. In Proc. 15th Int'l. Symp. on Experimental Algorithms, pages 150-166, 2016. URL: http://dx.doi.org/10.1007/978-3-319-38851-9_11.
http://dx.doi.org/10.1007/978-3-319-38851-9_11
L. Georgiadis, G. F. Italiano, L. Laura, and N. Parotsidis. 2-edge connectivity in directed graphs. In Proc. 26th ACM-SIAM Symp. on Discrete Algorithms, pages 1988-2005, 2015.
L. Georgiadis, G. F. Italiano, L. Laura, and N. Parotsidis. 2-vertex connectivity in directed graphs. In Proc. 42nd Int'l. Coll. on Automata, Languages, and Programming, pages 605-616, 2015.
L. Georgiadis, G. F. Italiano, C. Papadopoulos, and N. Parotsidis. Approximating the smallest spanning subgraph for 2-edge-connectivity in directed graphs. In ESA 2015, pages 582-594, 2015.
L. Georgiadis, G. F. Italiano, and N. Parotsidis. Incremental 2-edge connectivity in directed graphs. In Proc. 43rd Int'l. Coll. on Automata, Languages, and Programming, 2016. To appear.
L. Georgiadis, L. Laura, N. Parotsidis, and R. E. Tarjan. Loop nesting forests, dominators, and applications. In Proc. 13th Int'l. Symp. on Experimental Algorithms, pages 174-186, 2014.
Y. Guo, F. Kuipers, and P. Van Mieghem. Link-disjoint paths for reliable QoS routing. International Journal of Communication Systems, 16(9):779-798, 2003. URL: http://dx.doi.org/10.1002/dac.612.
http://dx.doi.org/10.1002/dac.612
M. Henzinger, V. King, and T. J. Warnow. Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica, 24(1):1-13, 1999. URL: http://dx.doi.org/10.1007/PL00009268.
http://dx.doi.org/10.1007/PL00009268
M. Henzinger, S. Krinninger, and V. Loitzenbauer. Finding 2-edge and 2-vertex strongly connected components in quadratic time. In Proc. 42nd Int'l. Coll. on Automata, Languages, and Programming, pages 713-724, 2015.
A. Itai and M. Rodeh. The multi-tree approach to reliability in distributed networks. Information and Computation, 79(1):43-59, 1988.
G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theor. Comput. Sci., 48(3):273-281, 1986. URL: http://dx.doi.org/10.1016/0304-3975(86)90098-8.
http://dx.doi.org/10.1016/0304-3975(86)90098-8
G. F. Italiano, L. Laura, and F. Santaroni. Finding strong bridges and strong articulation points in linear time. Theoretical Computer Science, 447:74-84, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2011.11.011.
http://dx.doi.org/10.1016/j.tcs.2011.11.011
R. Jaberi. Computing the 2-blocks of directed graphs. RAIRO-Theor. Inf. Appl., 49(2):93-119, 2015. URL: http://dx.doi.org/10.1051/ita/2015001.
http://dx.doi.org/10.1051/ita/2015001
R. Jaberi. On computing the 2-vertex-connected components of directed graphs. Discrete Applied Mathematics, 204:164-172, 2016. URL: http://dx.doi.org/10.1016/j.dam.2015.10.001.
http://dx.doi.org/10.1016/j.dam.2015.10.001
B. Laekhanukit, S. O. Gharan, and M. Singh. A rounding by sampling approach to the minimum size k-arc connected subgraph problem. In ICALP 2012, pages 606-616, 2012.
T. Lengauer and R. E. Tarjan. A fast algorithm for finding dominators in a flowgraph. ACM Transactions on Programming Languages and Systems, 1(1):121-41, 1979.
K. Menger. Zur allgemeinen kurventheorie. Fund. Math., 10:96-115, 1927.
H. Nagamochi and T. Ibaraki. Algorithmic Aspects of Graph Connectivity. Cambridge University Press, 2008. 1st edition.
H. Nagamochi and T. Watanabe. Computing k-edge-connected components of a multigraph. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E76-A(4):513-517, 1993.
R. E. Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972.
R. E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215-225, 1975.
J. Westbrook and R. E. Tarjan. Maintaining bridge-connected and biconnected components on-line. Algorithmica, 7(5&6):433-464, 1992. URL: http://dx.doi.org/10.1007/BF01758773.
http://dx.doi.org/10.1007/BF01758773
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Algorithms with Provable Guarantees for Clustering (Invited Talk)
In this talk, we give an overview of the current best approximation algorithms for fundamental clustering problems, such as k-center, k-median, k-means, and facility location. We focus on recent progress and point out several important open problems.
For the uncapacitated versions, a variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to a standard linear programming relaxation. This has given a uniform way of addressing these problems resulting in small constant approximation guarantees.
In spite of this impressive progress, it remains a challenging open problem to give tight guarantees. Moreover, this collection of powerful algorithmic techniques is not easily applicable to the capacitated setting. In fact, there is no simple strong convex relaxation known for the capacitated versions. As a result, our understanding of these problems is significantly weaker and several fundamental questions remain open.
Approximation algorithms
clustering
2:1-2:1
Invited Talk
Ola
Svensson
Ola Svensson
10.4230/LIPIcs.ESA.2016.2
Creative Commons Attribution 3.0 Unported license
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Beating Ratio 0.5 for Weighted Oblivious Matching Problems
We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem.
Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs).
(1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014).
(2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as \emph{matching coverage}, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied.
Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.
Weighted matching
oblivious algorithms
Ranking
linear programming
3:1-3:18
Regular Paper
Melika
Abolhassani
Melika Abolhassani
T.-H. Hubert
Chan
T.-H. Hubert Chan
Fei
Chen
Fei Chen
Hossein
Esfandiari
Hossein Esfandiari
MohammadTaghi
Hajiaghayi
MohammadTaghi Hajiaghayi
Mahini
Hamid
Mahini Hamid
Xiaowei
Wu
Xiaowei Wu
10.4230/LIPIcs.ESA.2016.3
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online vertex-weighted bipartite matching and single-bid budgeted allocations. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1253-1264, 2011.
Jonathan Aronson, Martin Dyer, Alan Frieze, and Stephen Suen. Randomized greedy matching. ii. Random Struct. Algorithms, 6(1):55-73, January 1995. URL: http://dx.doi.org/10.1002/rsa.3240060107.
http://dx.doi.org/10.1002/rsa.3240060107
Benjamin Birnbaum and Claire Mathieu. On-line bipartite matching made simple. SIGACT News, 39(1):80-87, March 2008. URL: http://dx.doi.org/10.1145/1360443.1360462.
http://dx.doi.org/10.1145/1360443.1360462
Niv Buchbinder, Kamal Jain, and Joseph Seffi Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In ESA, pages 253-264, 2007.
T.-H. Hubert Chan, Fei Chen, Xiaowei Wu, and Zhichao Zhao. Ranking on arbitrary graphs: Rematch via continuous lp with monotone and boundary condition constraints. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1112-1122, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.82.
http://dx.doi.org/10.1137/1.9781611973402.82
Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized primal-dual analysis of ranking for online bipartite matching. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 101-107, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.7.
http://dx.doi.org/10.1137/1.9781611973105.7
Martin E. Dyer and Alan M. Frieze. Randomized greedy matching. Random Struct. Algorithms, 2(1):29-46, 1991. URL: http://dx.doi.org/10.1002/rsa.3240020104.
http://dx.doi.org/10.1002/rsa.3240020104
Jon Feldman, Nitish Korula, Vahab S. Mirrokni, S. Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In Internet and Network Economics, 5th International Workshop, WINE 2009, Rome, Italy, December 14-18, 2009. Proceedings., pages 374-385, 2009. URL: http://dx.doi.org/10.1007/978-3-642-10841-9_34.
http://dx.doi.org/10.1007/978-3-642-10841-9_34
Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 982-991, 2008.
Gagan Goel and Pushkar Tripathi. Matching with our eyes closed. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 718-727, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.19.
http://dx.doi.org/10.1109/FOCS.2012.19
Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 587-596, 2011. URL: http://dx.doi.org/10.1145/1993636.1993715.
http://dx.doi.org/10.1145/1993636.1993715
Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 352-358, 1990. URL: http://dx.doi.org/10.1145/100216.100262.
http://dx.doi.org/10.1145/100216.100262
Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 597-606, 2011. URL: http://dx.doi.org/10.1145/1993636.1993716.
http://dx.doi.org/10.1145/1993636.1993716
Silvio Micali and Vijay V. Vazirani. An O(√V E) algorithm for finding maximum matching in general graphs. In FOCS'80, pages 17-27. IEEE Computer Society, 1980. URL: http://dx.doi.org/10.1109/SFCS.1980.12.
http://dx.doi.org/10.1109/SFCS.1980.12
Alvin E Roth, Tayfun Sönmez, and M Utku Ünver. Pairwise kidney exchange. Journal of Economic theory, 125(2):151-188, 2005.
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Outer Common Tangents and Nesting of Convex Hulls in Linear Time and Constant Workspace
We describe the first algorithm to compute the outer common tangents of two disjoint simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies on the same side of the line. An outer common tangent of two polygons is a tangent of both polygons such that the polygons lie on the same side of the tangent. Each polygon is given as a read-only array of its corners in cyclic order. The algorithm detects if an outer common tangent does not exist, which is the case if and only if the convex hull of one of the polygons is contained in the convex hull of the other. Otherwise, two corners defining an outer common tangent are returned.
simple polygon
common tangent
optimal algorithm
constant workspace
4:1-4:15
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Bartosz
Walczak
Bartosz Walczak
10.4230/LIPIcs.ESA.2016.4
M. Abrahamsen. An optimal algorithm computing edge-to-edge visibility in a simple polygon. In 25th Canadian Conference on Computational Geometry (CCCG 2013), pages 157-162, 2013.
M. Abrahamsen. An optimal algorithm for the separating common tangents of two polygons. In 31st International Symposium on Computational Geometry (SoCG 2015), volume 34 of LIPIcs, pages 198-208, 2015. http://arxiv.org/abs/1511.04036 (corrected version).
http://arxiv.org/abs/1511.04036
T. Asano, K. Buchin, M. Buchin, M. Korman, W. Mulzer, G. Rote, and A. Schulz. Memory-constrained algorithms for simple polygons. Comput. Geom., 46(8):959-969, 2013.
T. Asano, W. Mulzer, G. Rote, and Y. Wang. Constant-work-space algorithms for geometric problems. J. Comput. Geom., 2(1):46-68, 2011.
L. Barba, M. Korman, S. Langerman, K. Sadakane, and R.I. Silveira. Space-time trade-offs for stack-based algorithms. Algorithmica, 72(4):1097-1129, 2015.
L. Barba, M. Korman, S. Langerman, and R.I. Silveira. Computing the visibility polygon using few variables. Comput. Geom., 47(9):918-926, 2014.
G.S. Brodal and R. Jacob. Dynamic planar convex hull. In 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pages 617-626, 2002.
E. Carson, J. Demmel, L. Grigori, N. Knight, P. Koanantakool, O. Schwartz, and H.V. Simhadri. Write-avoiding algorithms. Technical Report http://www.eecs.berkeley.edu/Pubs/TechRpts/2015/EECS-2015-163.html, University of California, Berkeley, 2015.
http://www.eecs.berkeley.edu/Pubs/TechRpts/2015/EECS-2015-163.html
O. Darwish and A. Elmasry. Optimal time-space tradeoff for the 2D convex-hull problem. In European Symposium on Algorithms (ESA 2014), volume 8737 of LNCS, pages 284-295. Springer, 2014.
L. Guibas, J. Hershberger, and J. Snoeyink. Compact interval trees: a data structure for convex hulls. Int. J. Comput. Geom. Appl., 1(1):1-22, 1991.
S. Har-Peled. Shortest path in a polygon using sublinear space. In 31st International Symposium on Computational Geometry (SoCG 2015), volume 34 of LIPIcs, pages 111-125, 2015.
J. Hershberger and S. Suri. Applications of a semi-dynamic convex hull algorithm. BIT Numer. Math., 32(2):249-267, 1992.
D. Kirkpatrick and J. Snoeyink. Computing common tangents without a separating line. In 4th International Workshop on Algorithms and Data Structures (WADS 1995), volume 955 of LNCS, pages 183-193. Springer, 1995.
M. Korman, W. Mulzer, A. van Renssen, M. Roeloffzen, P. Seiferth, and Y. Stein. Time-space trade-offs for triangulations and voronoi diagrams. In Workshop on Algorithms and Data Structures (WADS 2015), volume 9214 of LNCS, pages 482-494. Springer, 2015.
A.A. Melkman. On-line construction of the convex hull of a simple polyline. Inform. Process. Lett., 25(1):11-12, 1987.
M.H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. System Sci., 23(2):166-204, 1981.
F.P. Preparata and S.J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20(2):87-93, 1977.
G.T. Toussaint. Solving geometric problems with the rotating calipers. In IEEE Mediterranean Electrotechnical Conference (MELECON 1983), pages A10.02/1-4, 1983.
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Sublinear Distance Labeling
A distance labeling scheme labels the n nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A D-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least D from each other. In this paper we consider distance labeling schemes for the classical case of unweighted and undirected graphs.
We present a O(n/D * log^2(D)) bit D-preserving distance labeling scheme, improving the previous bound by Bollobás et al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of Omega(n/D). With our D-preserving distance labeling scheme as a building block, we additionally achieve the following results:
1. We present the first distance labeling scheme of size o(n) for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require Omega(n) bits, Moon [Proc. of Glasgow Math. Association 1965].
2. For approximate r-additive labeling schemes, that return distances within an additive error of r we show a scheme of size
O(n/r * polylog(r*log(n))/log(n)) for r >= 2. This improves on the current best bound of O(n/r) by Alstrup et al. [SODA 2016] for sub-polynomial r, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for r=2.
Graph labeling schemes
Distance labeling
Graph theory
Sparse graphs
5:1-5:15
Regular Paper
Stephen
Alstrup
Stephen Alstrup
Søren
Dahlgaard
Søren Dahlgaard
Mathias Bæk Tejs
Knudsen
Mathias Bæk Tejs Knudsen
Ely
Porat
Ely Porat
10.4230/LIPIcs.ESA.2016.5
S. Abiteboul, H. Kaplan, and T. Milo. Compact labeling schemes for ancestor queries. In Proc. of the 12th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 547-556, 2001.
I. Abraham, S. Chechik, and C. Gavoille. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. In Proc. 44th Annual ACM Symp. on Theory of Computing (STOC), pages 1199-1218, 2012.
R. Agarwal, P. B. Godfrey, and S. Har-Peled. Approximate distance queries and compact routing in sparse graphs. In INFOCOM 2011. 30th IEEE International Conference on Computer Communications, pages 1754-1762, 2011.
T. Akiba, Y. Iwata, and Y. Yoshida. Fast exact shortest-path distance queries on large networks by pruned landmark labeling. In ACM International Conference on Management of Data (SIGMOD), pages 349-360, 2013. URL: http://dx.doi.org/10.1145/2463676.2465315.
http://dx.doi.org/10.1145/2463676.2465315
S. Alstrup, S. Dahlgaard, and M. B. T. Knudsen. Optimal induced universal graphs and labeling schemes for trees. In Proc. 56th Annual Symp. on Foundations of Computer Science (FOCS), 2015.
S. Alstrup, S. Dahlgaard, M. B. T. Knudsen, and E. Porat. Sublinear distance labeling for sparse graphs. CoRR, abs/1507.02618, 2015. URL: http://arxiv.org/abs/1507.02618.
http://arxiv.org/abs/1507.02618
S. Alstrup, C. Gavoille, E. B. Halvorsen, and H. Petersen. Simpler, faster and shorter labels for distances in graphs. In Proc. 27th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 338-350, 2016.
S. Alstrup, H. Kaplan, M. Thorup, and U. Zwick. Adjacency labeling schemes and induced-universal graphs. In Proc. of the 47th Annual ACM Symp. on Theory of Computing (STOC), 2015.
S. Alstrup and T. Rauhe. Small induced-universal graphs and compact implicit graph representations. In Proc. 43rd Annual Symp. on Foundations of Computer Science (FOCS), pages 53-62, 2002.
F. Bazzaro and C. Gavoille. Localized and compact data-structure for comparability graphs. Discrete Mathematics, 309(11):3465-3484, 2009. URL: http://dx.doi.org/10.1016/j.disc.2007.12.091.
http://dx.doi.org/10.1016/j.disc.2007.12.091
B. Bollobás, D. Coppersmith, and M. Elkin. Sparse distance preservers and additive spanners. SIAM J. Discrete Math., 19(4):1029-1055, 2005. See also SODA'03. URL: http://dx.doi.org/10.1137/S0895480103431046.
http://dx.doi.org/10.1137/S0895480103431046
M. A. Breuer. Coding the vertexes of a graph. IEEE Trans. on Information Theory, IT-12:148-153, 1966.
M. A. Breuer and J. Folkman. An unexpected result on coding vertices of a graph. J. of Mathemathical analysis and applications, 20:583-600, 1967.
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C. Gavoille, D. Peleg, S. Pérennes, and R. Raz. Distance labeling in graphs. J. of Algorithms, 53(1):85-112, 2004. See also SODA'01. URL: http://dx.doi.org/10.1016/j.jalgor.2004.05.002.
http://dx.doi.org/10.1016/j.jalgor.2004.05.002
P. Gawrychowski, A. Kosowski, and P. Uznanski. Even simpler distance labeling for (sparse) graphs. CoRR, abs/1507.06240, 2015. URL: http://arxiv.org/abs/1507.06240.
http://arxiv.org/abs/1507.06240
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A. Gupta, A. Kumar, and R. Rastogi. Traveling with a pez dispenser (or, routing issues in mpls). SIAM J. on Computing, 34(2):453-474, 2005. See also FOCS'01.
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M. Thorup. Compact oracles for reachability and approximate distances in planar digraphs. J. ACM, 51(6):993-1024, 2004. See also FOCS'01. URL: http://dx.doi.org/10.1145/1039488.1039493.
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M. Thorup and U. Zwick. Approximate distance oracles. J. of the ACM, 52(1):1-24, 2005. See also STOC'01.
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http://dx.doi.org/10.1007/BF02579350
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Probabilistic Routing for On-Street Parking Search
An estimated 30% of urban traffic is caused by search for parking spots [Shoup, 2005]. Suggesting routes along highly probable parking spots could reduce traffic. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.
parking search
on-street parking
probabilistic routing
constrained optimization
dataset
6:1-6:13
Regular Paper
Tobias
Arndt
Tobias Arndt
Danijar
Hafner
Danijar Hafner
Thomas
Kellermeier
Thomas Kellermeier
Simon
Krogmann
Simon Krogmann
Armin
Razmjou
Armin Razmjou
Martin S.
Krejca
Martin S. Krejca
Ralf
Rothenberger
Ralf Rothenberger
Tobias
Friedrich
Tobias Friedrich
10.4230/LIPIcs.ESA.2016.6
Alan Agresti and Brent A. Coull. Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2):119-126, 1998. URL: http://www.jstor.org/stable/2685469.
http://www.jstor.org/stable/2685469
C Richard Cassady and John E Kobza. A probabilistic approach to evaluate strategies for selecting a parking space. Transportation Science, 32(1):30-42, 1998.
Simon Evenepoel, Jan Van Ooteghem, Sofie Verbrugge, Didier Colle, and Mario Pickavet. On-street smart parking networks at a fraction of their cost: performance analysis of a sampling approach. Transactions on Emerging Telecommunications Technologies, 25(1):136-149, 2014.
Ming Hua and Jian Pei. Probabilistic path queries in road networks: Traffic uncertainty aware path selection. In Proceedings of the 13th International Conference on Extending Database Technology, pages 347-358, 2010. URL: http://dx.doi.org/10.1145/1739041.1739084.
http://dx.doi.org/10.1145/1739041.1739084
Gregor Jossé, Klaus Arthur Schmid, and Matthias Schubert. Probabilistic resource route queries with reappearance. In Proceedings of the 18th International Conference on Extending Database Technology, pages 445-456, 2015. URL: http://dx.doi.org/10.5441/002/edbt.2015.39.
http://dx.doi.org/10.5441/002/edbt.2015.39
Yaron Kanza, Eliyahu Safra, and Yehoshua Sagiv. Route search over probabilistic geospatial data. In Proceedings of the 11th International Symposium on Advances in Spatial and Temporal Databases, pages 153-170, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02982-0_12.
http://dx.doi.org/10.1007/978-3-642-02982-0_12
Eliyahu Safra, Yaron Kanza, Nir Dolev, Yehoshua Sagiv, and Yerach Doytsher. Computing a k-route over uncertain geographical data. In Proceedings of the 10th International Conference on Advances in Spatial and Temporal Databases, pages 276-293, 2007. URL: http://dl.acm.org/citation.cfm?id=1784462.1784478.
http://dl.acm.org/citation.cfm?id=1784462.1784478
Donald Shoup. The High Cost of Free Parking. APA Planners Press, 2005.
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Scalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths
Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.
isocontours
separating polygons
minimum-link paths
7:1-7:18
Regular Paper
Moritz
Baum
Moritz Baum
Thomas
Bläsius
Thomas Bläsius
Andreas
Gemsa
Andreas Gemsa
Ignaz
Rutter
Ignaz Rutter
Franziska
Wegner
Franziska Wegner
10.4230/LIPIcs.ESA.2016.7
Hannah Bast, Daniel Delling, Andrew V. Goldberg, Matthias Müller-Hannemann, Thomas Pajor, Peter Sanders, Dorothea Wagner, and Renato F. Werneck. Route Planning in Transportation Networks. Technical Report abs/1504.05140, ArXiv e-prints, 2015.
Veronika Bauer, Johann Gamper, Roberto Loperfido, Sylvia Profanter, Stefan Putzer, and Igor Timko. Computing Isochrones in Multi-Modal, Schedule-Based Transport Networks. In Proceedings of the 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (GIS'08), pages 78:1-78:2. ACM, 2008.
Moritz Baum, Thomas Bläsius, Andreas Gemsa, Ignaz Rutter, and Franziska Wegner. Scalable Isocontour Visualization in Road Networks via Minimum-Link Paths. Technical Report abs/1602.01777, ArXiv e-prints, 2016.
Moritz Baum, Valentin Buchhold, Julian Dibbelt, and Dorothea Wagner. Fast Exact Computation of Isochrones in Road Networks. In Proceedings of the 15th International Symposium on Experimental Algorithms (SEA'16), volume 9685 of Lecture Notes in Computer Science, pages 17-32. Springer, 2016.
Moritz Baum, Julian Dibbelt, Thomas Pajor, and Dorothea Wagner. Energy-Optimal Routes for Electric Vehicles. In Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (GIS'13), pages 54-63. ACM, 2013.
John L. Bentley and Thomas A. Ottmann. Algorithms for Reporting and Counting Geometric Intersections. IEEE Transactions on Computers, 28(9):643-647, 1979.
Francisc Bungiu, Michael Hemmer, John E. Hershberger, Kan Huang, and Alexander Kröller. Efficient Computation of Visibility Polygons. In Proceedings of the 30th European Workshop on Computational Geometry (EuroCG'14), 2014.
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Alexandros Efentakis, Sotiris Brakatsoulas, Nikos Grivas, Giorgos Lamprianidis, Kostas Patroumpas, and Dieter Pfoser. Towards a Flexible and Scalable Fleet Management Service. In Proceedings of the 6th ACM SIGSPATIAL International Workshop on Computational Transportation Science (IWTCS'13), pages 79-84. ACM, 2013.
Alexandros Efentakis, Nikos Grivas, George Lamprianidis, Georg Magenschab, and Dieter Pfoser. Isochrones, Traffic and DEMOgraphics. In Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (GIS'13), pages 548-551. ACM, 2013.
Alexandros Efentakis and Dieter Pfoser. GRASP. Extending Graph Separators for the Single-Source Shortest-Path Problem. In Proceedings of the 22nd Annual European Symposium on Algorithms (ESA'14), volume 8737 of Lecture Notes in Computer Science, pages 358-370. Springer, 2014.
Johann Gamper, Michael Böhlen, and Markus Innerebner. Scalable Computation of Isochrones with Network Expiration. In Proceedings of the 24th International Conference on Scientific and Statistical Database Management (SSDBM'12), volume 7338 of Lecture Notes in Computer Science, pages 526-543. Springer, 2012.
Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., 1979.
Ronald L. Graham. An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set. Information Processing Letters, 1(4):132-133, 1972.
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Leonidas J. Guibas, John E. Hershberger, Daniel Leven, Micha Sharir, and Robert E. Tarjan. Linear-Time Algorithms for Visibility and Shortest Path Problems Inside Triangulated Simple Polygons. Algorithmica, 2(1):209-233, 1987.
Leonidas J. Guibas, John E. Hershberger, Joseph S. B. Mitchell, and J. S. Snoeyink. Approximating Polygons and Subdivisions with Minimum-Link Paths. International Journal of Computational Geometry &Applications, 3(4):383-415, 1993.
Stefan Hausberger, Martin Rexeis, Michael Zallinger, and Raphael Luz. Emission Factors from the Model PHEM for the HBEFA Version 3. Technical Report I-20/2009, University of Technology, Graz, 2009.
Hiroshi Imai and Masao Iri. An Optimal Algorithm for Approximating a Piecewise Linear Function. Journal of Information Processing, 9(3):159-162, 1987.
Markus Innerebner, Michael Böhlen, and Johann Gamper. ISOGA: A System for Geographical Reachability Analysis. In Proceedings of the 12th International Conference on Web and Wireless Geographical Information Systems (W2GIS'13), volume 7820 of Lecture Notes in Computer Science, pages 180-189. Springer, 2013.
Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk, and Frank Staals. On the Complexity of Minimum-Link Path Problems. In Proceedings of the 32nd Annual Symposium on Computational Geometry (SoCG'16), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 49:1-49:16. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2016.
Sarunas Marciuska and Johann Gamper. Determining Objects within Isochrones in Spatial Network Databases. In Proceedings of the 14th East European Conference on Advances in Databases and Information Systems (ADBIS'10), volume 6295 of Lecture Notes in Computer Science, pages 392-405. Springer, 2010.
Joseph S. B. Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-Link Paths Revisited. Computational Geometry, 47(6):651-667, 2014.
Joseph S. B. Mitchell, Günter Rote, and Gerhard Woeginger. Minimum-Link Paths Among Obstacles in the Plane. Algorithmica, 8(1):431-459, 1992.
David O'Sullivan, Alastair Morrison, and John Shearer. Using Desktop GIS for the Investigation of Accessibility by Public Transport: An Isochrone Approach. International Journal of Geographical Information Science, 14(1):85-104, 2000.
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Robert E. Tarjan. Efficiency of a Good But Not Linear Set Union Algorithm. Journal of the ACM, 22(2):215-225, 1975.
Cao An Wang. Finding Minimal Nested Polygons. BIT Numerical Mathematics, 31(2):230-236, 1991.
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Computing Equilibria in Markets with Budget-Additive Utilities
We present the first analysis of Fisher markets with buyers that have budget-additive utility functions. Budget-additive utilities are elementary concave functions with numerous applications in online adword markets and revenue optimization problems. They extend the standard case of linear utilities and have been studied in a variety of other market models. In contrast to the frequently studied CES utilities, they have a global satiation point which can imply multiple market equilibria with quite different characteristics. Our main result is an efficient combinatorial algorithm to compute a market equilibrium with a Pareto-optimal allocation of goods. It relies on a new descending-price approach and, as a special case, also implies a novel combinatorial algorithm for computing a market equilibrium in linear Fisher markets. We complement this positive result with a number of hardness results for related computational questions. We prove that it isNP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.
Budget-Additive Utility
Market Equilibrium
Equilibrium Computation
8:1-8:14
Regular Paper
Xiaohui
Bei
Xiaohui Bei
Jugal
Garg
Jugal Garg
Martin
Hoefer
Martin Hoefer
Kurt
Mehlhorn
Kurt Mehlhorn
10.4230/LIPIcs.ESA.2016.8
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Lattice Distortion Problem
We introduce and study the Lattice Distortion Problem (LDP). LDP asks how "similar" two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal
distortion is one.
As our first contribution, we show that the distortion between any two lattices is approximated up to a n^{O(log(n))} factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a 2^{O(n*log(log(n))/log(n))} factor of optimal in polynomial time and within a n^{O(log(n))} factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.
lattices
lattice distortion
lattice isomoprhism
geometry of numbers
basis reduction
9:1-9:17
Regular Paper
Huck
Bennett
Huck Bennett
Daniel
Dadush
Daniel Dadush
Noah
Stephens-Davidowitz
Noah Stephens-Davidowitz
10.4230/LIPIcs.ESA.2016.9
Miklós Ajtai. Generating hard instances of lattice problems. In STOC, 1996.
Miklós Ajtai. The Shortest Vector Problem in L2 is NP-hard for randomized reductions. In STOC, 1998. URL: http://dx.doi.org/10.1145/276698.276705.
http://dx.doi.org/10.1145/276698.276705
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A sieve algorithm for the Shortest Lattice Vector Problem. In STOC, pages 601-610, 2001. URL: http://dx.doi.org/10.1145/380752.380857.
http://dx.doi.org/10.1145/380752.380857
Sanjeev Arora, Alan Frieze, and Haim Kaplan. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Mathematical Programming, 2002. URL: http://dx.doi.org/10.1007/s101070100271.
http://dx.doi.org/10.1007/s101070100271
Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Yadu Vasudev. Approximate Graph Isomorphism. In Mathematical Foundations of Computer Science, 2012.
L. Babai. Graph Isomorphism in quasipolynomial time, 2016. URL: http://arxiv.org/abs/1512.03547.
http://arxiv.org/abs/1512.03547
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http://dx.doi.org/10.1007/BF01445125
Zvika Brakerski and Vinod Vaikuntanathan. Lattice-based FHE as secure as PKE. In ITCS, 2014. URL: http://dx.doi.org/10.1145/2554797.2554799.
http://dx.doi.org/10.1145/2554797.2554799
J. Conway and N.J.A. Sloane. Sphere Packings, Lattices and Groups. Springer New York, 1998.
Nicolas Gama and Phong Q. Nguyen. Finding short lattice vectors within Mordell’s inequality. In STOC, 2008. URL: http://dx.doi.org/10.1145/1374376.1374408.
http://dx.doi.org/10.1145/1374376.1374408
Craig Gentry. Fully homomorphic encryption using ideal lattices. In STOC, 2009.
Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, 2008.
Oded Goldreich, Daniele Micciancio, Shmuel Safra, and Jean-Paul Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 71(2):55-61, 1999. URL: http://dx.doi.org/10.1016/S0020-0190(99)00083-6.
http://dx.doi.org/10.1016/S0020-0190(99)00083-6
Ishay Haviv and Oded Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. Theory of Computing, 8(23):513-531, 2012. Preliminary version in STOC'07.
Ishay Haviv and Oded Regev. On the Lattice Isomorphism Problem. In SODA, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.29.
http://dx.doi.org/10.1137/1.9781611973402.29
Ravi Kannan. Minkowski’s convex body theorem and Integer Programming. Mathematics of Operations Research, 12(3):pp. 415-440, 1987. URL: http://www.jstor.org/stable/3689974.
http://www.jstor.org/stable/3689974
Subhash Khot. Hardness of approximating the Shortest Vector Problem in lattices. Journal of the ACM, 52(5):789-808, September 2005. Preliminary version in FOCS'04.
A. Korkine and G. Zolotareff. Sur les formes quadratiques. Mathematische Annalen, 6(3):366-389, 1873. URL: http://dx.doi.org/10.1007/BF01442795.
http://dx.doi.org/10.1007/BF01442795
J. C. Lagarias, Hendrik W. Lenstra Jr., and Claus-Peter Schnorr. Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10(4):333-348, 1990. URL: http://dx.doi.org/10.1007/BF02128669.
http://dx.doi.org/10.1007/BF02128669
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http://dx.doi.org/10.1007/BF01457454
Hendrik W. Lenstra Jr. and Alice Silverberg. Lattices with symmetry, 2014. URL: http://arxiv.org/abs/1501.00178.
http://arxiv.org/abs/1501.00178
Daniele Micciancio. The Shortest Vector Problem is NP-hard to approximate to within some constant. SIAM Journal on Computing, 30(6):2008-2035, March 2001. Preliminary version in FOCS 1998.
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Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM J. Comput., 42(3):1364-1391, 2013. URL: http://dx.doi.org/10.1137/100811970.
http://dx.doi.org/10.1137/100811970
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Plurality Consensus in Arbitrary Graphs: Lessons Learned from Load Balancing
We consider plurality consensus in networks of n nodes. Initially, each node has one of k opinions. The nodes execute a (randomized) distributed protocol to agree on the plurality opinion (the opinion initially supported by the most nodes). In certain types of networks the nodes can be quite cheap and simple, and hence one seeks protocols that are not only time efficient but also simple and space efficient. Typically, protocols depend heavily on the employed communication mechanism, which ranges from sequential (only one pair of nodes communicates at any time) to fully parallel (all nodes communicate with all their neighbors at once) and everything in-between.
We propose a framework to design protocols for a multitude of communication mechanisms. We introduce protocols that solve the plurality consensus problem and are, with probability 1-o(1), both time and space efficient. Our protocols are based on an interesting relationship between plurality consensus and distributed load balancing. This relationship allows us to design protocols that generalize the state of the art for a large range of problem parameters.
Plurality Consensus
Distributed Computing
Load Balancing
10:1-10:18
Regular Paper
Petra
Berenbrink
Petra Berenbrink
Tom
Friedetzky
Tom Friedetzky
Peter
Kling
Peter Kling
Frederik
Mallmann-Trenn
Frederik Mallmann-Trenn
Chris
Wastell
Chris Wastell
10.4230/LIPIcs.ESA.2016.10
D. Aldous and J. Fill. Reversible markov chains and random walks on graphs, 2002. Unpublished. URL: http://www.stat.berkeley.edu/~aldous/RWG/book.html.
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Dan Alistarh, Rati Gelashvili, and Milan Vojnovic. Fast and exact majority in population protocols. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, (PODC), pages 47-56, 2015.
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Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale, and Riccardo Silvestri. Plurality consensus in the gossip model. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 371-390, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.27.
http://dx.doi.org/10.1137/1.9781611973730.27
Luca Becchetti, Andrea E. F. Clementi, Emanuele Natale, Francesco Pasquale, Riccardo Silvestri, and Luca Trevisan. Simple dynamics for plurality consensus. In 26th ACM Symposium on Parallelism in Algorithms and Architectures, (SPAA), pages 247-256, 2014. URL: http://dx.doi.org/10.1145/2612669.2612677.
http://dx.doi.org/10.1145/2612669.2612677
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Petra Berenbrink, Colin Cooper, Tom Friedetzky, Tobias Friedrich, and Thomas Sauerwald. Randomized diffusion for indivisible loads. J. Comput. Syst. Sci., 81(1):159-185, 2015.
Petra Berenbrink, Tom Friedetzky, George Giakkoupis, and Peter Kling. Efficient plurality consensus, or: The benefits of cleaning up from time to time. In Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP), 2016. to appear.
Stephen P. Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah. Randomized gossip algorithms. IEEE Transactions on Information Theory, 52(6):2508-2530, 2006.
Luca Cardelli and Attila Csikász-Nagy. The cell cycle switch computes approximate majority. Scientific reports, 2, 2012.
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Andrea E. F. Clementi, Miriam Di Ianni, Giorgio Gambosi, Emanuele Natale, and Riccardo Silvestri. Distributed community detection in dynamic graphs. Theor. Comput. Sci., 584:19-41, 2015.
Colin Cooper, Robert Elsässer, and Tomasz Radzik. The power of two choices in distributed voting. In Automata, Languages, and Programming - 41st International Colloquium, (ICALP), pages 435-446, 2014.
Colin Cooper, Robert Elsässer, Tomasz Radzik, Nicolas Rivera, and Takeharu Shiraga. Fast consensus for voting on general expander graphs. In Proceedings of the 29th International Symposium on Distributed Computing (DISC), pages 248-262, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48653-5_17.
http://dx.doi.org/10.1007/978-3-662-48653-5_17
Benjamin Doerr, Leslie Ann Goldberg, Lorenz Minder, Thomas Sauerwald, and Christian Scheideler. Stabilizing consensus with the power of two choices. In Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, (SPAA), pages 149-158, 2011.
Moez Draief and Milan Vojnovic. Convergence speed of binary interval consensus. SIAM J. Control and Optimization, 50(3):1087-1109, 2012.
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Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Frederik Mallmann-Trenn, and Horst Trinker. Efficient k-party voting with two choices. CoRR, abs/1602.04667, 2016. URL: http://arxiv.org/abs/1602.04667.
http://arxiv.org/abs/1602.04667
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George B. Mertzios, Sotiris E. Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Determining majority in networks with local interactions and very small local memory. In Automata, Languages, and Programming - 41st International Colloquium, (ICALP), pages 871-882, 2014.
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http://dx.doi.org/10.1007/s10458-013-9230-4
Elchanan Mossel and Grant Schoenebeck. Reaching consensus on social networks. In Innovations in Computer Science - (ICS), pages 214-229, 2010.
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http://arxiv.org/abs/1201.2715
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On the Hardness of Learning Sparse Parities
This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EventSet problem: given a homogeneous system of linear equations over $\F_2$ on $n$ variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EventSet has been a notorious open problem. Towards this goal, we show that unless \kclq can be solved in n^{o(k)} time, k-EventSet has no polynomial time algorithm when k = omega(log^2(n)).
Our work also shows that the non-homogeneous generalization of the problem - which we call k-VectorSum - is W[1]-hard on instances where the number of equations is O(k*log(n)), improving on previous reductions which produced Omega(n) equations. We use the hardness of k-VectorSum as a starting point to prove the result for k-EventSet, and additionally strengthen the former to show the hardness of approximately learning k-juntas. In particular, we prove that given a system of O(exp(O(k))*log(n)) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or any function on at most k-variables (a k-junta) satisfies at most (1/2 + epsilon)-fraction of the equations, for any constant epsilon > 0. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities.
In a similar vein, we use the hardness of k-EventSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time, there is no poly(m,n)*2^{o(sqrt{k})} time algorithm to decide whether a given set of $m$ points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx Pr_{F_2^n}[P({z}) = 0] fraction of the points i.e., P is fooled by the set of points.
Lastly, we study the approximation in the sparsity of the solution. Let the Gap-k-VectorSum problem be: given an instance of k-VectorSum of size n, decide if there exist a k-sparse solution, or every solution is of sparsity at least k' = (1+delta_0)k. Assuming the Exponential Time Hypothesis, we show that for some constants c_0, delta_0 > 0 there is no poly(n) time algorithm for Gap-k-VectorSum when k = omega((log(log( n)))^{c_0}).
Fixed Parameter Tractable
Juntas
Minimum Distance of Code
Psuedorandom Generators
11:1-11:17
Regular Paper
Arnab
Bhattacharyya
Arnab Bhattacharyya
Ameet
Gadekar
Ameet Gadekar
Suprovat
Ghoshal
Suprovat Ghoshal
Rishi
Saket
Rishi Saket
10.4230/LIPIcs.ESA.2016.11
Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In Proc. 22nd Annual European Symposium on Algorithms, pages 1-12. Springer, 2014.
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Vikraman Arvind, Johannes Köbler, and Wolfgang Lindner. Parameterized learnability of juntas. Theor. Comp. Sci., 410(47-49):4928-4936, 2009.
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Arnab Bhattacharyya, Ameet Gadekar, Suprovat Ghoshal, and Rishi Saket. On the hardness of learning sparse parities. CoRR, abs/1511.08270, 2015. URL: http://arxiv.org/abs/1511.08270.
http://arxiv.org/abs/1511.08270
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Online Algorithms for Multi-Level Aggregation
In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests.
Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting.
algorithmic aspects of networks
online algorithms
scheduling and resource allocation
12:1-12:17
Regular Paper
Marcin
Bienkowski
Marcin Bienkowski
Martin
Böhm
Martin Böhm
Jaroslaw
Byrka
Jaroslaw Byrka
Marek
Chrobak
Marek Chrobak
Christoph
Dürr
Christoph Dürr
Lukas
Folwarczny
Lukas Folwarczny
Lukasz
Jez
Lukasz Jez
Jiri
Sgall
Jiri Sgall
Nguyen
Kim Thang
Nguyen Kim Thang
Pavel
Vesely
Pavel Vesely
10.4230/LIPIcs.ESA.2016.12
Alok Aggarwal and James K. Park. Improved algorithms for economic lot sizing problems. Operations Research, 41:549-571, 1993.
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Marcin Bienkowski, Jaroslaw Byrka, Marek Chrobak, Neil B. Dobbs, Tomasz Nowicki, Maxim Sviridenko, Grzegorz Swirszcz, and Neal E. Young. Approximation algorithms for the joint replenishment problem with deadlines. Journal of Scheduling, 18(6):545-560, 2015.
Marcin Bienkowski, Jaroslaw Byrka, Marek Chrobak, Łukasz Jeż, Dorian Nogneng, and Jiří Sgall. Better approximation bounds for the joint replenishment problem. In Proc. 25th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 42-54, 2014.
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Niv Buchbinder, Tracy Kimbrel, Retsef Levi, Konstantin Makarychev, and Maxim Sviridenko. Online make-to-order joint replenishment model: Primal-dual competitive algorithms. In Proc. 19th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 952-961, 2008.
Niv Buchbinder and Joseph (Seffi) Naor. The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009.
Marek Chrobak, Claire Kenyon, John Noga, and Neal E. Young. Incremental medians via online bidding. Algorithmica, 50(4):455-478, 2008.
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Retsef Levi, Robin Roundy, and David B. Shmoys. A constant approximation algorithm for the one-warehouse multi-retailer problem. In Proc. 16th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 365-374, 2005.
Retsef Levi, Robin Roundy, and David B. Shmoys. Primal-dual algorithms for deterministic inventory problems. Mathematics of Operations Research, 31(2):267-284, 2006.
Retsef Levi, Robin Roundy, David B. Shmoys, and Maxim Sviridenko. A constant approximation algorithm for the one-warehouse multiretailer problem. Management Science, 54(4):763-776, 2008.
Retsef Levi and Maxim Sviridenko. Improved approximation algorithm for the one-warehouse multi-retailer problem. In Proc. 9th Int. Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX), pages 188-199, 2006.
Tim Nonner and Alexander Souza. Approximating the joint replenishment problem with deadlines. Discrete Mathematics, Algorithms and Applications, 1(2):153-174, 2009.
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Compact and Fast Sensitivity Oracles for Single-Source Distances
Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.
fault-tolerant shortest-path tree
approximate distance
distance sensitivity oracle
13:1-13:14
Regular Paper
Davide
Bilo
Davide Bilo
Luciano
Guala
Luciano Guala
Stefano
Leucci
Stefano Leucci
Guido
Proietti
Guido Proietti
10.4230/LIPIcs.ESA.2016.13
Surender Baswana and Neelesh Khanna. Approximate shortest paths avoiding a failed vertex: Near optimal data structures for undirected unweighted graphs. Algorithmica, 66(1):18-50, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9621-y.
http://dx.doi.org/10.1007/s00453-012-9621-y
Michael A. Bender and Martin Farach-Colton. The level ancestor problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004. URL: http://dx.doi.org/10.1016/j.tcs.2003.05.002.
http://dx.doi.org/10.1016/j.tcs.2003.05.002
Omer Berkman and Uzi Vishkin. Finding level-ancestors in trees. J. Comput. Syst. Sci., 48(2):214-230, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80002-9.
http://dx.doi.org/10.1016/S0022-0000(05)80002-9
Aaron Bernstein and David R. Karger. A nearly optimal oracle for avoiding failed vertices and edges. In STOC, pages 101-110, 2009.
Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Fault-tolerant approximate shortest-path trees. In ESA, pages 137-148, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_12.
http://dx.doi.org/10.1007/978-3-662-44777-2_12
Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-edge-fault-tolerant approximate shortest-path trees. In STACS, pages 18:1-18:14, 2016.
Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-sensitivity distance oracles and routing schemes. In ESA, pages 84-96, 2010.
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Efficient Algorithms with Asymmetric Read and Write Costs
In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fundamentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We consider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M,omega)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost omega >> 1.
For FFT and sorting networks we show a lower bound cost of Omega(omega*n*log_{omega*M}(n)), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when omega is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(omega,log(n)/log(omega*M)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n*n diamond DAG of Omega(omega*n^2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. However, we show that for the minimum edit distance problem (and related problems), which would seem to be a diamond DAG, we can beat this lower bound with an algorithm with only O(omega*n^2/(M*min(omega^{1/3},M^{1/2}))) cost. To achieve this we make use of a "path sketch" technique that is forbidden in a strict DAG computation. Finally, we show several interesting upper bounds for shortest path problems, minimum spanning trees, and other problems. A common theme in many of the upper bounds is that they require redundant computation and a tradeoff between reads and writes.
Computational Model
Lower Bounds
Shortest-paths
Non-Volatile Memory
Sorting Networks
Fast Fourier Transform
Diamond DAG
Minimum Spanning Tree
14:1-14:18
Regular Paper
Guy E.
Blelloch
Guy E. Blelloch
Jeremy T.
Fineman
Jeremy T. Fineman
Phillip B.
Gibbons
Phillip B. Gibbons
Yan
Gu
Yan Gu
Julian
Shun
Julian Shun
10.4230/LIPIcs.ESA.2016.14
Alok Aggarwal and Jeffrey S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9), 1988. URL: http://dx.doi.org/10.1145/48529.48535.
http://dx.doi.org/10.1145/48529.48535
Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA, 1974.
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Hyperbolic Random Graphs: Separators and Treewidth
Hyperbolic random graphs share many common properties with complex real-world networks; e.g., small diameter and average distance, large clustering coefficient, and a power-law degree sequence with adjustable exponent beta. Thus, when analyzing algorithms for large networks, potentially more realistic results can be achieved by assuming the input to be a hyperbolic random graph of size n. The worst-case run-time is then replaced by the expected run-time or by bounds that hold with high probability (whp), i.e., with probability 1-O(1/n). Though many structural properties of hyperbolic random graphs have been studied, almost no algorithmic results are known.
Divide-and-conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that they can be expected to have balanced separator hierarchies with separators of size O(n^{3/2-beta/2}), O(log n), and O(1) if 2 < beta < 3, beta = 3, and 3 < beta, respectively. We infer that these graphs have whp a treewidth of O(n^{3/2-beta/2}), O(log^2 n), and O(log n), respectively. For 2 < \beta < 3, this matches a known lower bound.
To demonstrate the usefulness of our results, we give several algorithmic applications.
hyperbolic random graphs
scale-free networks
power-law graphs
separators
treewidth
15:1-15:16
Regular Paper
Thomas
Bläsius
Thomas Bläsius
Tobias
Friedrich
Tobias Friedrich
Anton
Krohmer
Anton Krohmer
10.4230/LIPIcs.ESA.2016.15
Mohammed Amin Abdullah, Michel Bode, and Nikolaos Fountoulakis. Typical distances in a geometric model for complex networks. CoRR, abs/1506.07811:1-33, 2015.
Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999.
Michael Bode, Nikolaos Fountoulakis, and Tobias Müller. On the largest component of a hyperbolic model of complex networks. The Electronic Journal of Combinatorics, 22(3):1-46, 2015.
Marián Boguñá, Fragkiskos Papadopoulos, and Dmitri Krioukov. Sustaining the internet with hyperbolic mapping. Nature Communications, 1(62), 2010.
Béla Bollobás and Oliver M. Riordan. Mathematical Results on Scale-Free Random Graphs, chapter 1, pages 1-34. Wiley, 2005.
Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. CoRR, abs/1511.00576:1-42, 2015.
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Tobias Friedrich and Anton Krohmer. Cliques in hyperbolic random graphs. In Proceedings of the IEEE Conference on Computer Communications (INFOCOM'15), pages 1544-1552, 2015.
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Anshui Li and Tobias Müller. On the treewidth of random geometric graphs and percolated grids. Manuscript, 2015.
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Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane
Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of Omega(n^2). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.
hyperbolic random graphs
embedding
power-law graphs
hyperbolic plane
16:1-16:18
Regular Paper
Thomas
Bläsius
Thomas Bläsius
Tobias
Friedrich
Tobias Friedrich
Anton
Krohmer
Anton Krohmer
Sören
Laue
Sören Laue
10.4230/LIPIcs.ESA.2016.16
William Aiello, Fan Chung, and Linyuan Lu. A random graph model for massive graphs. In 32nd Symp. Theory of Computing (STOC), pages 171-180, 2000.
William Aiello, Fan Chung, and Linyuan Lu. A random graph model for power law graphs. Experimental Mathematics, 10(1):53-66, 2001.
Rodrigo Aldecoa, Chiara Orsini, and Dmitri Krioukov. Hyperbolic graph generator. Computer Physics Communications, 196:492-496, 2015. URL: http://dx.doi.org/10.1016/j.cpc.2015.05.028.
http://dx.doi.org/10.1016/j.cpc.2015.05.028
Dena Marie Asta and Cosma Rohilla Shalizi. Geometric network comparisons. In 31st Conference on Uncertainty in Artificial Intelligence (UAI), pages 102-110, 2015.
Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286:509-512, 1999.
Michel Bode, Nikolaos Fountoulakis, and Tobias Müller. On the giant component of random hyperbolic graphs. In 7th European Conf. Combinatorics, Graph Theory and Applications, pages 425-429, 2013.
Michel Bode, Nikolaos Fountoulakis, and Tobias Müller. The probability that the hyperbolic random graph is connected. https://www.math.uu.nl/~Muell001/Papers/BFM.pdf, 2014.
https://www.math.uu.nl/~Muell001/Papers/BFM.pdf
Marián Boguñá, Fragkiskos Papadopoulos, and Dmitri Krioukov. Sustaining the internet with hyperbolic mapping. Nature Communications, 1:62, 2010.
Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. arXiv preprint arXiv:1511.00576, 2015.
F. Chung and L. Lu. Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6(2):125-145, 2002.
Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman. Power-law distributions in empirical data. SIAM Review, 51(4):661-703, 2009.
James R. Clough and Tim S. Evans. Embedding graphs in lorentzian spacetime. arXiv 1602.03103, 2016.
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Tobias Friedrich and Anton Krohmer. On the diameter of hyperbolic random graphs. In 42nd Intl. Coll. Automata, Languages and Programming (ICALP), pages 614-625, 2015.
Luca Gugelmann, Konstantinos Panagiotou, and Ueli Peter. Random hyperbolic graphs: degree sequence and clustering. In 39th Intl. Coll. Automata, Languages and Programming (ICALP), pages 573-585, 2012.
Stephen G. Kobourov. Handbook of Graph Drawing and Visualization, chapter Force-Directed Drawing Algorithms, pages 383-408. Chapman and Hall/CRC, 2013.
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Christoph Koch and Johannes Lengler. Bootstrap percolation on geometric inhomogeneous random graphs. In 43rd Intl. Coll. Automata, Languages and Programming (ICALP), 2016.
Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. Hyperbolic geometry of complex networks. Phys. Rev. E, 82:036106, 2010.
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Creative Commons Attribution 3.0 Unported license
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Fully Dynamic Spanners with Worst-Case Update Time
An alpha-spanner of a graph G is a subgraph H such that H preserves all distances of G within a factor of alpha. In this paper, we give fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update. In particular, our algorithms maintain:
- a 3-spanner with ~O(n^{1+1/2}) edges with worst-case update time ~O(n^{3/4}), or
- a 5-spanner with ~O(n^{1+1/3}) edges with worst-case update time ~O (n^{5/9}).
These size/stretch tradeoffs are best possible (up to logarithmic factors). They can be extended to the weighted setting at very minor cost. Our algorithms are randomized and correct with high probability against an oblivious adversary. We also further extend our techniques to construct a 5-spanner with suboptimal size/stretch tradeoff, but improved worst-case update time.
To the best of our knowledge, these are the first dynamic spanner algorithms with sublinear worst-case update time guarantees. Since it is known how to maintain a spanner using small amortized}but large worst-case update time [Baswana et al. SODA'08], obtaining algorithms with strong worst-case bounds, as presented in this paper, seems to be the next natural step for this problem.
Dynamic graph algorithms
spanners
17:1-17:18
Regular Paper
Greg
Bodwin
Greg Bodwin
Sebastian
Krinninger
Sebastian Krinninger
10.4230/LIPIcs.ESA.2016.17
Ittai Abraham and Shiri Chechik. Dynamic decremental approximate distance oracles with (1+ε,2) stretch. CoRR, abs/1307.1516, 2013. URL: http://arxiv.org/abs/1307.1516.
http://arxiv.org/abs/1307.1516
Ittai Abraham, Shiri Chechik, and Kunal Talwar. Fully dynamic all-pairs shortest paths: Breaking the o(n) barrier. In International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 1-16, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.1.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.1
Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. CoRR, abs/1604.02094, 2015. URL: http://arxiv.org/abs/1604.02094.
http://arxiv.org/abs/1604.02094
Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, and B. Riva Shalom. Online dictionary matching with one gap. CoRR, abs/1503.07563, 2015. URL: http://arxiv.org/abs/1503.07563.
http://arxiv.org/abs/1503.07563
Giorgio Ausiello, Paolo Giulio Franciosa, and Giuseppe F. Italiano. Small stretch spanners on dynamic graphs. Journal of Graph Algorithms and Applications, 10(2):365-385, 2006. Announced at ESA'05. URL: http://dx.doi.org/10.7155/jgaa.00133.
http://dx.doi.org/10.7155/jgaa.00133
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http://dx.doi.org/10.1145/4221.4227
Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments, chapter Arbdefective Coloring. Morgan and Claypool, 2013.
Surender Baswana. Streaming algorithm for graph spanners - single pass and constant processing time per edge. Information Processing Letters, 106(3):110-114, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2007.11.001.
http://dx.doi.org/10.1016/j.ipl.2007.11.001
Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM Journal on Computing, 39(7):2865-2896, 2010. Announced at FOCS'10. URL: http://dx.doi.org/10.1137/080737174.
http://dx.doi.org/10.1137/080737174
Surender Baswana, Sumeet Khurana, and Soumojit Sarkar. Fully dynamic randomized algorithms for graph spanners. ACM Transactions on Algorithms, 8(4):35:1-35:51, 2012. Announced at ESA'06, and SODA'08. URL: http://dx.doi.org/10.1145/2344422.2344425.
http://dx.doi.org/10.1145/2344422.2344425
Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures &Algorithms, 30(4):532-563, 2007. Announced at ICALP'03. URL: http://dx.doi.org/10.1002/rsa.20130.
http://dx.doi.org/10.1002/rsa.20130
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http://dx.doi.org/10.1137/1.9781611973082.104
Shiri Chechik. Approximate distance oracles with constant query time. In Symposium on Theory of Computing (STOC), pages 654-663, 2014. URL: http://dx.doi.org/10.1145/2591796.2591801.
http://dx.doi.org/10.1145/2591796.2591801
Shiri Chechik. Approximate distance oracles with improved bounds. In Symposium on Theory of Computing (STOC), pages 1-10, 2015. URL: http://dx.doi.org/10.1145/2746539.2746562.
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Lenore Cowen. Compact routing with minimum stretch. Journal of Algorithms, 38(1):170-183, 2001. Announced at SODA'99. URL: http://dx.doi.org/10.1006/jagm.2000.1134.
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Michael Elkin. Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. ACM Transactions on Algorithms, 7(2):20:1-20:17, 2011. Announced at ICALP'07. URL: http://dx.doi.org/10.1145/1921659.1921666.
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http://dx.doi.org/10.1109/FOCS.2014.24
Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. A subquadratic-time algorithm for dynamic single-source shortest paths. In Symposium on Discrete Algorithms (SODA), pages 1053-1072, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.79.
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http://dx.doi.org/10.1137/090776573
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http://dx.doi.org/10.1016/0095-8956(91)90097-4
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fixed-Parameter Approximability of Boolean MinCSPs
The minimum unsatisfiability version of a constraint satisfaction problem (CSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size set of constraints whose deletion makes the instance satisfiable. For a finite set Gamma of constraints, we denote by CSP(Gamma) the restriction of the problem where each constraint is from Gamma. The polynomial-time solvability and the polynomial-time approximability of CSP(Gamma) were fully characterized by [Khanna et al. SICOMP 2000]. Here we study the fixed-parameter (FP-) approximability of the problem: given an instance and an integer k, one has to find a solution of size at most g(k) in time f(k)n^{O(1)} if a solution of size at most k exists. We especially focus on the case of constant-factor FP-approximability. Our main result classifies each finite constraint language Gamma into one of three classes: (1) CSP(Gamma) has a constant-factor FP-approximation; (2) CSP(Gamma) has a (constant-factor) FP-approximation if and only if Nearest Codeword has a (constant-factor) FP-approximation; (3) CSP(Gamma) has no FP-approximation, unless FPT=W[P]. We show that problems in the second class do not have constant-factor FP-approximations if both the Exponential-Time Hypothesis (ETH) and the Linear PCP Conjecture (LPC) hold. We also show that such an approximation would imply the existence of an FP-approximation for the k-Densest Subgraph problem with ratio 1-epsilon for any epsilon>0.
constraint satisfaction problems
approximability
fixed-parameter tractability
18:1-18:18
Regular Paper
Édouard
Bonnet
Édouard Bonnet
László
Egri
László Egri
Dániel
Marx
Dániel Marx
10.4230/LIPIcs.ESA.2016.18
Eric Allender, Michael Bauland, Neil Immerman, Henning Schnoor, and Heribert Vollmer. The complexity of satisfiability problems: Refining Schaefer’s theorem. J. Comput. Syst. Sci., 75(4):245-254, 2009.
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http://dx.doi.org/10.1006/jcss.1997.1472
Arnab Bhattacharyya and Yuichi Yoshida. An algebraic characterization of testable boolean CSPs. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 123-134, 2013.
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Elmar Böhler, Steffen Reith, Henning Schnoor, and Heribert Vollmer. Bases for boolean co-clones. Inf. Process. Lett., 96(2):59-66, 2005.
Édouard Bonnet, László Egri, and Dániel Marx. Fixed-parameter approximability of boolean minCSPs. CoRR, abs/1601.04935, 2016. URL: http://arxiv.org/abs/1601.04935.
http://arxiv.org/abs/1601.04935
Édouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On subexponential and fpt-time inapproximability. Algorithmica, 71(3):541-565, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9889-1.
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Andrei A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
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Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. J. ACM, 60(5):34, 2013. URL: http://dx.doi.org/10.1145/2528400.
http://dx.doi.org/10.1145/2528400
Andrei A. Bulatov, Martin E. Dyer, Leslie Ann Goldberg, Markus Jalsenius, Mark Jerrum, and David Richerby. The complexity of weighted and unweighted #CSP. J. Comput. Syst. Sci., 78(2):681-688, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.12.002.
http://dx.doi.org/10.1016/j.jcss.2011.12.002
Andrei A. Bulatov and Dániel Marx. The complexity of global cardinality constraints. Logical Methods in Computer Science, 6(4), 2010. URL: http://dx.doi.org/10.2168/LMCS-6(4:4)2010.
http://dx.doi.org/10.2168/LMCS-6(4:4)2010
Andrei A. Bulatov and Dániel Marx. Constraint satisfaction parameterized by solution size. SIAM J. Comput., 43(2):573-616, 2014. URL: http://dx.doi.org/10.1137/120882160.
http://dx.doi.org/10.1137/120882160
Liming Cai and Xiuzhen Huang. Fixed-parameter approximation: Conceptual framework and approximability results. Algorithmica, 57(2):398-412, 2010. URL: http://dx.doi.org/10.1007/s00453-008-9223-x.
http://dx.doi.org/10.1007/s00453-008-9223-x
Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94-114, 2006. URL: http://dx.doi.org/10.1007/s00037-006-0210-9.
http://dx.doi.org/10.1007/s00037-006-0210-9
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http://eccc.hpi-web.de/eccc-reports/2007/TR07-106/index.html
Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Guy Kortsarz. Fixed-parameter and approximation algorithms: A new look. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 110-122, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03898-8_11.
http://dx.doi.org/10.1007/978-3-319-03898-8_11
Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity Classifications of Boolean Constraint Satisfaction Problems. Society for Industrial and Applied Mathematics, 2001.
Nadia Creignou and Heribert Vollmer. Parameterized complexity of weighted satisfiability problems: Decision, enumeration, counting. Fundam. Inform., 136(4):297-316, 2015.
Pierluigi Crescenzi and Alessandro Panconesi. Completeness in approximation classes. Inf. Comput., 93(2):241-262, 1991. URL: http://dx.doi.org/10.1016/0890-5401(91)90025-W.
http://dx.doi.org/10.1016/0890-5401(91)90025-W
Robert Crowston, Gregory Gutin, Mark Jones, and Anders Yeo. Parameterized complexity of satisfying almost all linear equations over 𝔽₂. Theory Comput. Syst., 52(4):719-728, 2013. URL: http://dx.doi.org/10.1007/s00224-012-9415-2.
http://dx.doi.org/10.1007/s00224-012-9415-2
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Víctor Dalmau and Andrei A. Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. TOCT, 5(4):15, 2013. URL: http://dx.doi.org/10.1145/2540090.
http://dx.doi.org/10.1145/2540090
Víctor Dalmau, Andrei A. Krokhin, and Rajsekar Manokaran. Towards a characterization of constant-factor approximable min csps. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 847-857, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.58.
http://dx.doi.org/10.1137/1.9781611973730.58
Vladimir G. Deineko, Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. The approximability of MAXCSP with fixed-value constraints. J. ACM, 55(4), 2008. URL: http://dx.doi.org/10.1145/1391289.1391290.
http://dx.doi.org/10.1145/1391289.1391290
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
http://dx.doi.org/10.1007/978-1-4471-5559-1
Rodney G. Downey, Michael R. Fellows, Catherine McCartin, and Frances Rosamond. Parameterized approximation of dominating set problems. Inform. Process. Lett., 109(1):68-70, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.09.017.
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J. Flum and M. Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 2006.
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Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. The approximability of three-valued MAXCSP. SIAM J. Comput., 35(6):1329-1349, 2006. URL: http://dx.doi.org/10.1137/S009753970444644X.
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Vladimir Kolmogorov and Stanislav Zivny. The complexity of conservative valued CSPs. J. ACM, 60(2):10, 2013. URL: http://dx.doi.org/10.1145/2450142.2450146.
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Stefan Kratsch, Dániel Marx, and Magnus Wahlström. Parameterized complexity and kernelizability of MaxOnes and ExactOnes problems. In Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings, pages 489-500, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15155-2_43.
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Stefan Kratsch and Magnus Wahlström. Preprocessing of MinOnes problems: A dichotomy. In Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, pages 653-665, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14165-2_55.
http://dx.doi.org/10.1007/978-3-642-14165-2_55
Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11(2):15:1-15:31, 2014. URL: http://dx.doi.org/10.1145/2566616.
http://dx.doi.org/10.1145/2566616
Dániel Marx. Parameterized complexity of constraint satisfaction problems. Computational Complexity, 14(2):153-183, 2005. URL: http://dx.doi.org/10.1007/s00037-005-0195-9.
http://dx.doi.org/10.1007/s00037-005-0195-9
Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008.
Dániel Marx. Completely inapproximable monotone and antimonotone parameterized problems. J. Comput. Syst. Sci., 79(1):144-151, 2013.
Dániel Marx and Igor Razgon. Constant ratio fixed-parameter approximation of the edge multicut problem. Inf. Process. Lett., 109(20):1161-1166, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2009.07.016.
http://dx.doi.org/10.1016/j.ipl.2009.07.016
Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. J. ACM, 57(5), 2010. URL: http://dx.doi.org/10.1145/1754399.1754402.
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R. Pöschel and Kalužnin. Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin, 1979.
Emil L. Post. On the Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press, 1941.
Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, 2009.
Thomas J. Schaefer. The complexity of satisfiability problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, Calif., 1978), pages 216-226. ACM, New York, 1978.
Johan Thapper and Stanislav Zivny. The complexity of finite-valued CSPs. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 695-704, 2013. URL: http://dx.doi.org/10.1145/2488608.2488697.
http://dx.doi.org/10.1145/2488608.2488697
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Parameterized Hardness of Art Gallery Problems
Given a simple polygon P on n vertices, two points x,y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S.
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out a f(k)*n^{o(k/log k)} algorithm, for any computable function f, where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n^{O(k)} algorithms that exist for both problems.
art gallery problem
computational geometry
parameterized complexity
ETH-based lower bound
geometric set cover/hitting set
19:1-19:17
Regular Paper
Édouard
Bonnet
Édouard Bonnet
Tillmann
Miltzow
Tillmann Miltzow
10.4230/LIPIcs.ESA.2016.19
Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: http://dx.doi.org/10.1016/j.jalgor.2003.10.001.
http://dx.doi.org/10.1016/j.jalgor.2003.10.001
Saugata Basu, Richard Pollack, and Marie-Francoise Roy. Algorithms in real algebraic geometry. AMC, 10:12, 2011.
Édouard Bonnet and Tillmann Miltzow. The parameterized hardness of the art gallery problem. CoRR, abs/1603.08116, 2016. URL: http://arxiv.org/abs/1603.08116.
http://arxiv.org/abs/1603.08116
Björn Brodén, Mikael Hammar, and Bengt J. Nilsson. Guarding lines and 2-link polygons is apx-hard. In Proceedings of the 13th Canadian Conference on Computational Geometry, University of Waterloo, Ontario, Canada, August 13-15, 2001, pages 45-48, 2001. URL: https://dspace.mah.se/handle/2043/6645.
https://dspace.mah.se/handle/2043/6645
Sergio Cabello, Panos Giannopoulos, Christian Knauer, Dániel Marx, and Günter Rote. Geometric clustering: Fixed-parameter tractability and lower bounds with respect to the dimension. ACM Trans. Algorithms, 7(4):43, 2011. URL: http://dx.doi.org/10.1145/2000807.2000811.
http://dx.doi.org/10.1145/2000807.2000811
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Mark de Berg, Hans Bodlaender, and Sándor Kisfaludi-Bak. Connected dominating set in unit-disk graphs is w[1]-hard. In EuroCG 2016, 2016.
Pedro Jussieu de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. CoRR, abs/1410.8720, 2014. URL: http://arxiv.org/abs/1410.8720.
http://arxiv.org/abs/1410.8720
Rodney G. Downey and Michael R. Fellows. Parameterized complexity. Springer Science &Business Media, 2012.
Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.05.014.
http://dx.doi.org/10.1016/j.ipl.2006.05.014
Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001.
Steve Fisk. A short proof of Chvátal’s watchman theorem. J. Comb. Theory, Ser. B, 24(3):374, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90059-X.
http://dx.doi.org/10.1016/0095-8956(78)90059-X
Jörg Flum and Martin Grohe. Parameterized complexity theory, volume xiv of texts in theoretical computer science. an EATCS series, 2006.
Subir K. Ghosh. Visibility algorithms in the plane. Cambridge University Press, 2007.
Subir K. Ghosh. Approximation algorithms for art gallery problems in polygons. Discrete Applied Mathematics, 158(6):718-722, 2010.
Panos Giannopoulos and Christian Knauer. Finding a largest empty convex subset in space is w[1]-hard. CoRR, abs/1304.0247, 2013. URL: http://arxiv.org/abs/1304.0247.
http://arxiv.org/abs/1304.0247
Panos Giannopoulos, Christian Knauer, and Günter Rote. The parameterized complexity of some geometric problems in unbounded dimension. In Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers, pages 198-209, 2009. URL: http://dx.doi.org/10.1007/978-3-642-11269-0_16.
http://dx.doi.org/10.1007/978-3-642-11269-0_16
Panos Giannopoulos, Christian Knauer, and Sue Whitesides. Parameterized complexity of geometric problems. Comput. J., 51(3):372-384, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm053.
http://dx.doi.org/10.1093/comjnl/bxm053
Matt Gibson, Erik Krohn, and Qing Wang. The VC-dimension of visibility on the boundary of a simple polygon. In Algorithms and Computation, pages 541-551. Springer, 2015.
Alexander Gilbers and Rolf Klein. A new upper bound for the VC-dimension of visibility regions. Computational Geometry, 47(1):61-74, 2014.
Russell Impagliazzo and Ramamohan Paturi. Complexity of k-sat. In Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on, pages 237-240. IEEE, 1999.
Gil Kalai and Jiří Matoušek. Guarding galleries where every point sees a large area. Israel Journal of Mathematics, 101(1):125-139, 1997.
Matthew J. Katz and Gabriel S. Roisman. On guarding the vertices of rectilinear domains. Computational Geometry, 39(3):219-228, 2008.
James King. Fast vertex guarding for polygons with and without holes. Comput. Geom., 46(3):219-231, 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2012.07.004.
http://dx.doi.org/10.1016/j.comgeo.2012.07.004
David G. Kirkpatrick. An O(lg lg OPT)-approximation algorithm for multi-guarding galleries. Discrete & Computational Geometry, 53(2):327-343, 2015. URL: http://dx.doi.org/10.1007/s00454-014-9656-8.
http://dx.doi.org/10.1007/s00454-014-9656-8
Erik Krohn and Bengt J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9653-3.
http://dx.doi.org/10.1007/s00453-012-9653-3
Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 154-165, 2006. URL: http://dx.doi.org/10.1007/11847250_14.
http://dx.doi.org/10.1007/11847250_14
Dániel Marx. Can you beat treewidth? Theory of Computing, 6(1):85-112, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a005.
http://dx.doi.org/10.4086/toc.2010.v006a005
Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. In ESA 2015, pages 865-877, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
http://dx.doi.org/10.1007/978-3-662-48350-3_72
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http://dx.doi.org/10.1016/0022-0000(90)90017-F
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Pavel Valtr. Guarding galleries where no point sees a small area. Israel Journal of Mathematics, 104(1):1-16, 1998.
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KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation
We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks.
The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest.
The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E|^{1/2+o(1)} with high probability, obtaining a significant speedup with respect to the Theta(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well.
The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well.
Betweenness centrality
shortest path algorithm
graph mining
sampling
network analysis
20:1-20:18
Regular Paper
Michele
Borassi
Michele Borassi
Emanuele
Natale
Emanuele Natale
10.4230/LIPIcs.ESA.2016.20
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, apsp and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1681-1697. SIAM, 2015.
Amir Abboud, Virginia V. Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter. In Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-391, may 2016. URL: http://arxiv.org/abs/1506.0179.
http://arxiv.org/abs/1506.0179
Ankit Aggarwal, Amit Deshpande, and Ravi Kannan. Adaptive Sampling for k-Means Clustering. In Irit Dinur, Klaus Jansen, Joseph Naor, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, number 5687 in Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2009.
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David A. Bader, Shiva Kintali, Kamesh Madduri, and Milena Mihail. Approximating betweenness centrality. The 5th Workshop on Algorithms and Models for the Web-Graph, 2007.
Alex Bavelas. A mathematical model for group structures. Human organization, 7(3):16-30, 1948.
Elisabetta Bergamini. private communication, 2016.
Elisabetta Bergamini and Henning Meyerhenke. Fully-dynamic approximation of betweenness centrality. In ESA, 2015.
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http://dx.doi.org/10.1016/S0195-6698(80)80030-8
Michele Borassi, Pierluig Crescenzi, and Michel Habib. Into the square - On the complexity of some quadratic-time solvable problems. In Proceedings of the 16th Italian Conference on Theoretical Computer Science (ICTCS), pages 1-17, 2015.
Michele Borassi, Pierluigi Crescenzi, and Luca Trevisan. An Axiomatic and an Average-Case Analysis of Algorithms and Heuristics for Metric Properties of Graphs. arXiv:1604.01445 [cs], April 2016.
Michele Borassi and Emanuele Natale. Kadabra is an adaptive algorithm for betweenness via random approximation. arXiv preprint arXiv:1604.08553, 2016.
Stephen P. Borgatti and Martin G. Everett. A graph-theoretic perspective on centrality. Social Networks, 28:466-484, 2006.
Ulrik Brandes. A faster algorithm for betweenness centrality. The Journal of Mathematical Sociology, 25(2):163-177, jun 2001. URL: http://dx.doi.org/10.1080/0022250X.2001.9990249.
http://dx.doi.org/10.1080/0022250X.2001.9990249
Ulrik Brandes. On variants of shortest-path betweenness centrality and their generic computation. Social Networks, 30:136-145, 2008.
Ulrik Brandes and Christian Pich. Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos, 17(07):2303-2318, 2007. URL: http://dx.doi.org/10.1142/S0218127407018403.
http://dx.doi.org/10.1142/S0218127407018403
Bernard S Cohn and McKim Marriott. Networks and centres of integration in indian civilization. Journal of social Research, 1(1):1-9, 1958.
Shlomi Dolev, Yuval Elovici, and Rami Puzis. Routing betweenness centrality. J. ACM, 57, 2010.
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David Eppstein and Joseph Wang. Fast approximation of centrality. J. Graph Algorithms Appl., 8:39-45, 2001.
Dóra Erdős, Vatche Ishakian, Azer Bestavros, and Evimaria Terzi. A divide-and-conquer algorithm for betweenness centrality. In Proceedings of the 2015 SIAM International Conference on Data Mining, pages 433-441, 2015.
Robert Geisberger, Peter Sanders, Dominik Schultes, and Daniel Delling. Contraction hierarchies: Faster and simpler hierarchical routing in road networks. In Catherine C. McGeoch, editor, Experimental Algorithms: 7th International Workshop, WEA 2008, pages 319-333. Springer Berlin Heidelberg, 2008.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have Strongly Exponential Complexity? Journal of Computer and System Sciences, 63(4):512-530, dec 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Riko Jacob, Dirk Koschützki, Katharina Anna Lehmann, Leon Peeters, and Dagmar Tenfelde-Podehl. Algorithms for centrality indices. In DAGSTUHL, 2004.
Hermann Kaindl and Gerhard Kainz. Bidirectional heuristic search reconsidered. J. Artif. Intell. Res. (JAIR), 7:283-317, 1997.
Yeon-sup Lim, Daniel S Menasché, Bruno Ribeiro, Don Towsley, and Prithwish Basu. Online estimating the k central nodes of a network. Proceedings of IEEE NSW, pages 118-122, 2011.
Richard J. Lipton and Jeffrey F. Naughton. Query Size Estimation by Adaptive Sampling. Journal of Computer and System Sciences, 51(1):18-25, August 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1050.
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Linyuan Lu and Fan R. K. Chung. Complex graphs and networks. Number no. 107 in CBMS regional conference series in mathematics. American Mathematical Society, 2006.
Mark Newman. Networks: an introduction. OUP Oxford, 2010.
Mark EJ Newman. Scientific collaboration networks. ii. shortest paths, weighted networks, and centrality. Physical review E, 64(1):016132, 2001.
Ilkka Norros and Hannu Reittu. On a conditionally Poissonian graph process. Advances in Applied Probability, 38(1):59-75, 2006.
Jürgen Pfeffer and Kathleen M Carley. k-centralities: local approximations of global measures based on shortest paths. In Proceedings of the 21st international conference companion on World Wide Web, pages 1043-1050. ACM, 2012.
Andrea Pietracaprina, Matteo Riondato, Eli Upfal, and Fabio Vandin. Mining Top-K Frequent Itemsets Through Progressive Sampling. Data Mining and Knowledge Discovery, 21(2):310-326, September 2010. URL: http://dx.doi.org/10.1007/s10618-010-0185-7.
http://dx.doi.org/10.1007/s10618-010-0185-7
Ira Pohl. Bi-directional and heuristic search in path problems. PhD thesis, Dept. of Computer Science, Stanford University., 1969.
Matteo Riondato and Evgenios M Kornaropoulos. Fast approximation of betweenness centrality through sampling. Data Mining and Knowledge Discovery, 30(2):438-475, 2015.
Matteo Riondato and Eli Upfal. ABRA: Approximating Betweenness Centrality in Static and Dynamic Graphs with Rademacher Averages. arXiv preprint 1602.05866, pages 1-27, 2016. URL: http://arxiv.org/abs/1602.05866.
http://arxiv.org/abs/1602.05866
Ahmet Erdem Sariyüce, Erik Saule, Kamer Kaya, and Ümit V Çatalyürek. Shattering and compressing networks for betweenness centrality. In SIAM Data Mining Conference (SDM). SIAM, 2013.
Marvin E Shaw. Group structure and the behavior of individuals in small groups. The Journal of psychology, 38(1):139-149, 1954.
Alfonso Shimbel. Structural parameters of communication networks. The bulletin of mathematical biophysics, 15(4):501-507, 1953.
Christian L. Staudt, Aleksejs Sazonovs, and Henning Meyerhenke. Networkit: an interactive tool suite for high-performance network analysis. arXiv preprint 1403.3005, pages 1-25, 2014.
Remco van der Hofstad. Random graphs and complex networks. Vol. II. Manuscript, 2014.
Flavio Vella, Giancarlo Carbone, and Massimo Bernaschi. Algorithms and heuristics for scalable betweenness centrality computation on multi-gpu systems. CoRR, abs/1602.00963, 2016.
Stanley Wasserman and Katherine Faust. Social network analysis: Methods and applications, volume 8. Cambridge university press, 1994.
Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1867-1877, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.135.
http://dx.doi.org/10.1137/1.9781611973402.135
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Separation of Cycle Inequalities for the Periodic Timetabling Problem
Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem. We give the first pseudo-polynomial time separation algorithm for cycle inequalities, and we give a rigorous proof for the pseudo-polynomial time separability of the change-cycle inequalities. The efficiency of these cutting planes is demonstrated on real-world instances of the periodic timetabling problem.
periodic timetabling
cycle inequalities
separation algorithm
21:1-21:13
Regular Paper
Ralf
Borndörfer
Ralf Borndörfer
Heide
Hoppmann
Heide Hoppmann
Marika
Karbstein
Marika Karbstein
10.4230/LIPIcs.ESA.2016.21
Michael Bussieck. Gams - lop.gms: Line optimization. URL: http://www.gams.com/modlib/libhtml/lop.htm.
http://www.gams.com/modlib/libhtml/lop.htm
M. Kümmling, P. Großmann, K. Nachtigall, J. Opitz, and R. Weiß. A state-of-the-art realization of cyclic railway timetable computation. Public Transport, 7(3):281-293, 2015. URL: http://dx.doi.org/10.1007/s12469-015-0108-5.
http://dx.doi.org/10.1007/s12469-015-0108-5
Christian Liebchen. Periodic timetable optimization in public transport. PhD thesis, Technische Universtität Berlin, 2006.
Christian Liebchen and Rolf H. Möhring. The modeling power of the periodic event scheduling problem: Railway timetables - and beyond. In Frank Geraets, Leo Kroon, Anita Schoebel, Dorothea Wagner, and Christos D. Zaroliagis, editors, Algorithmic Methods for Railway Optimization, volume 4359 of Lecture Notes in Computer Science, pages 3-40. Springer Berlin Heidelberg, 2007.
Christian Liebchen and Leon Peeters. Integral cycle bases for cyclic timetabling. Discrete Optimization, 6:98-109, 2009.
Christian Liebchen and Elmar Swarat. The Second Chvatal Closure Can Yield Better Railway Timetables. In Matteo Fischetti and Peter Widmayer, editors, 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08), volume 9 of OpenAccess Series in Informatics (OASIcs), Dagstuhl, Germany, 2008. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Thomas Lindner. Train Schedule Optimization in Public Rail Transport. PhD thesis, Technische Universtität Braunschweig, 2000.
K. Nachtigall and J. Opitz. Solving periodic timetable optimisation problems by modulo simplex calculations. In Matteo Fischetti and Peter Widmayer, editors, ATMOS'08, volume 9, 2008.
Karl Nachtigall. Periodic Network Optimization and Fixed Interval Timetables. Habilitation thesis, Universtität Hildesheim, 1998.
Michiel A. Odijk. Construction of periodic timetables, part 1: A cutting plane algorithm. Technical Report 94-61, TU Delft, 1994.
Leon W. P. Peeters. Cyclic Railway and Timetable Optimization. PhD thesis, Erasmus Universiteit Rotterdam, 2003.
Alexander Schrijver. Routing and timetabling by topological search. Documenta Mathematica, Extra Volume ICM 1998:1-9, 1998.
P. Sels, T. Dewilde, D. Cattrysse, and P. Vansteenwegen. Reducing the passenger travel time in practice by the automated construction of a robust railway timetable. Transportation Research Part B: Methodological, 84:124-156, 2016. URL: http://dx.doi.org/10.1016/j.trb.2015.12.007.
http://dx.doi.org/10.1016/j.trb.2015.12.007
Paolo Serafini and Walter Ukovich. A mathematical model for periodic scheduling problems. SIAM Journal on Discrete Mathematics, 2(4):550-581, 1989.
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Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance
We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.
grid mapping
Hausdorff distance
Fréchet distance
digital geometry
22:1-22:16
Regular Paper
Quirijn W.
Bouts
Quirijn W. Bouts
Irina
Irina Kostitsyna
Irina Irina Kostitsyna
Marc
van Kreveld
Marc van Kreveld
Wouter
Meulemans
Wouter Meulemans
Willem
Sonke
Willem Sonke
Kevin
Verbeek
Kevin Verbeek
10.4230/LIPIcs.ESA.2016.22
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Hitting Set for Hypergraphs of Low VC-dimension
We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small.
VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014).
In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time.
To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.
hitting set
VC-dimension
23:1-23:18
Regular Paper
Karl
Bringmann
Karl Bringmann
László
Kozma
László Kozma
Shay
Moran
Shay Moran
N. S.
Narayanaswamy
N. S. Narayanaswamy
10.4230/LIPIcs.ESA.2016.23
Pankaj K. Agarwal and Jiangwei Pan. Near-linear algorithms for geometric hitting sets and set covers. In 30th Annual Symposium on Computational Geometry, SOCG'14, Kyoto, Japan, June 08 - 11, 2014, page 271, 2014. URL: http://dx.doi.org/10.1145/2582112.2582152.
http://dx.doi.org/10.1145/2582112.2582152
Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014. URL: http://eccc.hpi-web.de/report/2014/135.
http://eccc.hpi-web.de/report/2014/135
Noga Alon, Dana Moshkovitz, and Shmuel Safra. Algorithmic construction of sets for k-restrictions. ACM Transactions on Algorithms, 2(2):153-177, 2006. URL: http://dx.doi.org/10.1145/1150334.1150336.
http://dx.doi.org/10.1145/1150334.1150336
Anselm Blumer, A. Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the Vapnik-Chervonenkis Dimension. J. ACM, 36(4):929-965, October 1989. URL: http://dx.doi.org/10.1145/76359.76371.
http://dx.doi.org/10.1145/76359.76371
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http://dx.doi.org/10.1007/s00454-006-1273-8
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Panos Giannopoulos, Christian Knauer, and Sue Whitesides. Parameterized complexity of geometric problems. Comput. J., 51(3):372-384, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm053.
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Shay Moran, Amir Shpilka, Avi Wigderson, and Amir Yehudayoff. Teaching and compressing for low vc-dimension. Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, 2015.
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New Algorithms, Better Bounds, and a Novel Model for Online Stochastic Matching
Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1/epsilon) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including: (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to [Haeupler, Mirrokni and Zadimoghaddam WINE 2011] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of [Jaillet and Lu Math. Oper. Res 2013] to 0.7299. We also consider two extensions, one is "known I.I.D." with non-integral arrival rate and stochastic rewards; the other is "known I.I.D." b-matching with non-integral arrival rate and stochastic rewards. We present a simple non-adaptive algorithm which works well simultaneously on the two extensions.
One of the key ingredients of our improvement is the following (offline) approach to bipartite-matching polytopes with additional constraints. We first add several valid constraints in order to get a good fractional solution f; however, these give us less control over the structure of f. We next remove all these additional constraints and randomly move from f to a feasible point on the matching polytope with all coordinates being from the set {0, 1/k, 2/k,..., 1} for a chosen integer k. The structure of this solution is inspired by [Jaillet and Lu Math. Oper. Res 2013] and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately [exactly] with high probability [in expectation]). This underlies some of our improvements, and, we hope, could be of independent interest.
Ad-Allocation
Online Matching
Randomized Algorithms
24:1-24:16
Regular Paper
Brian
Brubach
Brian Brubach
Karthik Abinav
Sankararaman
Karthik Abinav Sankararaman
Aravind
Srinivasan
Aravind Srinivasan
Pan
Xu
Pan Xu
10.4230/LIPIcs.ESA.2016.24
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online vertex-weighted bipartite matching and single-bid budgeted allocations. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1253-1264. SIAM, 2011.
Saeed Alaei, MohammadTaghi Hajiaghayi, and Vahid Liaghat. Online prophet-inequality matching with applications to ad allocation. In Proceedings of the 13th ACM Conference on Electronic Commerce, pages 18-35. ACM, 2012.
Saeed Alaei, MohammadTaghi Hajiaghayi, and Vahid Liaghat. The online stochastic generalized assignment problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX, and 17th International Workshop, RANDOM, pages 11-25. Springer Berlin Heidelberg, 2013.
Bahman Bahmani and Michael Kapralov. Improved bounds for online stochastic matching. In European Symposium on Algorithms (ESA), pages 170-181. Springer, 2010.
Nikhil R Devanur and Thomas P Hayes. The adwords problem: online keyword matching with budgeted bidders under random permutations. In Proceedings of the 10th ACM conference on Electronic commerce, pages 71-78. ACM, 2009.
Nikhil R. Devanur, Kamal Jain, Balasubramanian Sivan, and Christopher A. Wilkens. Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In Proceedings of the 12th ACM Conference on Electronic Commerce, pages 29-38. ACM, 2011.
Nikhil R Devanur, Balasubramanian Sivan, and Yossi Azar. Asymptotically optimal algorithm for stochastic adwords. In Proceedings of the 13th ACM Conference on Electronic Commerce, pages 388-404. ACM, 2012.
Jon Feldman, Nitish Korula, Vahab Mirrokni, S Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In Internet and network economics, pages 374-385. Springer, 2009.
Jon Feldman, Aranyak Mehta, Vahab Mirrokni, and S Muthukrishnan. Online stochastic matching: Beating 1-1/e. In Foundations of Computer Science (FOCS), pages 117-126. IEEE, 2009.
Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3):324-360, 2006.
Bernhard Haeupler, Vahab S. Mirrokni, and Morteza Zadimoghaddam. Online stochastic weighted matching: Improved approximation algorithms. In Internet and Network Economics, volume 7090 of Lecture Notes in Computer Science, pages 170-181. Springer Berlin Heidelberg, 2011.
Patrick Jaillet and Xin Lu. Online stochastic matching: New algorithms with better bounds. Mathematics of Operations Research, 39(3):624-646, 2013.
Bala Kalyanasundaram and Kirk R Pruhs. An optimal deterministic algorithm for online b-matching. Theoretical Computer Science, 233(1):319-325, 2000.
Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 352-358. ACM, 1990.
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An optimal online algorithm for weighted bipartite matching and extensions to combinatorial auctions. In European Symposium on Algorithms (ESA), pages 589-600. Springer, 2013.
Nitish Korula and Martin Pál. Algorithms for secretary problems on graphs and hypergraphs. In Automata, Languages and Programming, pages 508-520. Springer, 2009.
Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 597-606. ACM, 2011.
Vahideh H Manshadi, Shayan Oveis Gharan, and Amin Saberi. Online stochastic matching: Online actions based on offline statistics. Mathematics of Operations Research, 37(4):559-573, 2012.
Aranyak Mehta. Online matching and ad allocation. Foundations and Trends in Theoretical Computer Science, 8(4):265-368, 2012.
Aranyak Mehta and Debmalya Panigrahi. Online matching with stochastic rewards. In Foundations of Computer Science (FOCS), pages 728-737. IEEE, 2012.
Aranyak Mehta, Bo Waggoner, and Morteza Zadimoghaddam. Online stochastic matching with unequal probabilities. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2015.
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Solving k-SUM Using Few Linear Queries
The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c<d where d is the ceiling of k/2. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n^3 log^2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in ~O(n^{d+8}) time, and performs O(n^3 log^2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n^3 log^2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-a-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P.
We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth ~O(n^3) for the
k-SUM problem.
k-SUM problem
linear decision trees
point location
$varepsilon$-nets
25:1-25:17
Regular Paper
Jean
Cardinal
Jean Cardinal
John
Iacono
John Iacono
Aurélien
Ooms
Aurélien Ooms
10.4230/LIPIcs.ESA.2016.25
Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms (ESA 2014), pages 1-12. Springer, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In International Colloquium on Automata, Languages, and Programming (ICALP 2014), volume 8572 of Lecture Notes in Computer Science, pages 39-51. Springer, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Symposium on Theory of Computing (STOC 2015), pages 41-50, 2015.
Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. J. ACM, 52(2):157-171, 2005.
Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In International Colloquium on Automata, Languages, and Programming (ICALP 2014), volume 8572 of Lecture Notes in Computer Science, pages 114-125. Springer, 2014.
Ilya Baran, Erik D. Demaine, and Mihai Patrascu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584-596, 2008.
Gill Barequet and Sariel Har-Peled. Polygon-containment and translational min-hausdorff-distance between segment sets are 3SUM-hard. In Symposium on Discrete Algorithms (SODA 1999), pages 862-863, 1999.
Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the vapnik-chervonenkis dimension. J. ACM, 36(4):929-965, 1989.
Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi. Algebraic complexity theory, volume 315 of Grundlehren der mathematischen Wissenschaften. Springer, 1997.
Jean Cardinal, Samuel Fiorini, Gwenaël Joret, Raphaël M Jungers, and J Ian Munro. An efficient algorithm for partial order production. SIAM journal on computing, 39(7):2927-2940, 2010.
Jean Cardinal, Samuel Fiorini, Gwenaël Joret, Raphaël M Jungers, and J Ian Munro. Sorting under partial information (without the ellipsoid algorithm). Combinatorica, 33(6):655-697, 2013.
Marco Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mikhailin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. Electronic Colloquium on Computational Complexity (ECCC 2015), 22:148, 2015.
Timothy M. Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Symposium on Theory of Computing (STOC 2015), pages 31-40. ACM, 2015.
Kenneth L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17(4):830-847, 1988.
David P. Dobkin and Richard J. Lipton. On some generalizations of binary search. In Symposium on Theory of Computing (STOC 1974), pages 310-316, 1974.
David P. Dobkin and Richard J. Lipton. A lower bound of the 1/2 n² on linear search programs for the knapsack problem. J. Comput. Syst. Sci., 16(3):413-417, 1978.
Jeff Erickson. Lower bounds for linear satisfiability problems. Chicago Journal of Theoretical Computer Science, 1999.
Hervé Fournier. Complexité et expressibilité sur les réels. PhD thesis, École normale supérieure de Lyon, 2001.
Ari Freund. Improved subquadratic 3SUM. Algorithmica, 2015. To appear.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Comput. Geom., 5:165-185, 1995.
O. Gold and M. Sharir. Improved bounds for 3SUM, k-SUM, and linear degeneracy. ArXiv e-prints, 2015. URL: http://arxiv.org/abs/1512.05279.
http://arxiv.org/abs/1512.05279
Jacob E. Goodman and Joseph O'Rourke, editors. Handbook of Discrete and Computational Geometry, Second Edition. Chapman and Hall/CRC, 2004.
Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science (FOCS 2014), pages 621-630. IEEE, 2014.
David Haussler and Emo Welzl. ε-nets and simplex range queries. Discrete &Computational Geometry, 2(1):127-151, 1987.
T. Kopelowitz, S. Pettie, and E. Porat. Higher lower bounds from the 3SUM conjecture. ArXiv e-prints, 2014. URL: http://arxiv.org/abs/1407.6756.
http://arxiv.org/abs/1407.6756
S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286-303, 1993.
Friedhelm Meyer auf der Heide. A polynomial linear search algorithm for the n-dimensional knapsack problem. J. ACM, 31(3):668-676, 1984.
Joseph S. B. Mitchell and Joseph O'Rourke. Computational geometry column 42. Int. J. Comput. Geometry Appl., 11(5):573-582, 2001.
Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Symposium on Theory of Computing (STOC 2010), pages 603-610, 2010.
Mihai Patrascu and Ryan Williams. On the possibility of faster SAT algorithms. In Symposium on Discrete Algorithms (SODA 2010), pages 1065-1075, 2010.
William L. Steiger and Ileana Streinu. A pseudo-algorithmic separation of lines from pseudo-lines. Information Processing Letters, 53(5):295-299, 1995.
Andrew Chi-Chih Yao. On parallel computation for the knapsack problem. J. ACM, 29(3):898-903, 1982.
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Optimal Staged Self-Assembly of General Shapes
We analyze the number of stages, tiles, and bins needed to construct n * n squares and scaled shapes in the staged tile assembly model. In particular, we prove that there exists a staged system with b bins and t tile types assembling an n * n square using O((log n - tb - t log t)/b^2 + log log b/log t) stages and Omega((log n - tb - t log t)/b^2) are necessary for almost all n. For a shape S, we prove O((K(S) - tb - t log t)/b^2 + (log log b)/log t) stages suffice and Omega((K(S) - tb - t log t)/b^2) are necessary for the assembly of a scaled version of S, where K(S) denotes the Kolmogorov complexity of S. Similarly tight bounds are also obtained when more powerful flexible glue functions are permitted. These are the first staged results that hold for all choices of b and t and generalize prior results.
The upper bound constructions use a new technique for efficiently converting each both sources of system complexity, namely the tile types and mixing graph, into a "bit string" assembly.
Tile self-assembly
2HAM
aTAM
DNA computing
biocomputing
26:1-26:17
Regular Paper
Cameron
Chalk
Cameron Chalk
Eric
Martinez
Eric Martinez
Robert
Schweller
Robert Schweller
Luis
Vega
Luis Vega
Andrew
Winslow
Andrew Winslow
Tim
Wylie
Tim Wylie
10.4230/LIPIcs.ESA.2016.26
Zachary Abel, Nadia Benbernou, Mirela Damian, Erik Demaine, Martin Demaine, Robin Flatland, Scott Kominers, and Robert Schweller. Shape replication through self-assembly and RNAse enzymes. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2010.
Leonard Adleman, Qi Cheng, Ashish Goel, and Ming-Deh Huang. Running time and program size for self-assembled squares. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 740-748, 2001.
Bahar Behsaz, Ján Maňuch, and Ladislav Stacho. Turing universality of step-wise and stage assembly at temperature 1. In DNA Computing and Molecular Programming (DNA), volume 7433 of LNCS, pages 1-11. Springer, 2012.
Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert Schweller, Scott M. Summers, and Andrew Winslow. Two hands are better than one (up to constant factors): Self-assembly in the 2HAM vs. aTAM. In Proceedings of 30th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 20 of LIPIcs, pages 172-184. Schloss Dagstuhl, 2013.
Ho-Lin Chen and David Doty. Parallelism and time in hierarchical self-assembly. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1163-1182, 2012.
Qi Cheng, Gagan Aggarwal, Michael H. Goldwasser, Ming-Yang Kao, Robert T. Schweller, and Pablo Moisset de Espanés. Complexities for generalized models of self-assembly. SIAM Journal on Computing, 34:1493-1515, 2005.
E. D. Demaine, M. J. Patitz, T. A. Rogers, R. T. Schweller, and D. Woods. The two-handed tile assembly model is not intrinsically universal. In Automata, Languages and Programming (ICALP), volume 7965 of LNCS, pages 400-412. Springer, 2013.
Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Mashhood Ishaque, Eynat Rafalin, Robert T. Schweller, and Diane L. Souvaine. Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Natural Computing, 7(3):347-370, 2008.
Erik D. Demaine, Sarah Eisenstat, Mashhood Ishaque, and Andrew Winslow. One-dimensional staged self-assembly. Natural Computing, 12(2):247-258, 2013.
Erik D. Demaine, Sándor P. Fekete, Christian Scheffer, and Arne Schmidt. New geometric algorithms for fully connected staged self-assembly. In DNA Computing and Molecular Programming (DNA), volume 9211 of LNCS, pages 104-116. Springer, 2015.
Erik D. Demaine, Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers. Self-assembly of arbitrary shapes using RNAse enzymes: Meeting the Kolmogorov bound with small scale factor (extended abstract). In Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 9 of LIPIcs, pages 201-212. Schloss Dagstuhl, 2011.
David Doty. Producibility in hierarchical self-assembly. In Proceedings of Unconventional Computation and Natural Computation (UCNC) 2014, pages 142-154, 2014.
Constantine Evans. Crystals that Count! Physical Principles and Experimental Investigations of DNA Tile Self-Assembly. PhD thesis, Caltech, 2014.
David Furcy, Samuel Micka, and Scott M. Summers. Optimal program-size complexity for self-assembly at temperature 1 in 3D. In DNA Computing and Molecular Programming (DNA), volume 9211 of LNCS, pages 71-86. Springer, 2015.
Yonggang Ke, Luvena L. Ong, William M. Shih, and Peng Yin. Three-dimensional structures self-assembled from dna bricks. Science, 338(6111):1177-1183, 2012.
Thomas H. Labean, Sung Ha Park, Sang Jung Ahn, and John H. Reif. Stepwise DNA self-assembly of fixed-size nanostructures. In Foundations of Nanoscience, Self-assembled Architectures, and Devices, pages 179-181, 2005.
Ján Maňuch, Ladislav Stacho, and Christine Stoll. Step-wise tile assembly with a constant number of tile types. Natural Computing, 11(3):535-550, 2012.
Matthew J. Patitz and Scott M. Summers. Identifying shapes using self-assembly. Algorithmica, 64:481-510, 2012.
Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC), pages 459-468, 2000.
David Soloveichik and Erik Winfree. Complexity of self-assembled shapes. SIAM Journal on Computing, 36(6):1544-1569, 2007.
Erik Winfree. Algorithmic Self-Assembly of DNA. PhD thesis, Caltech, 1998.
Andrew Winslow. Staged self-assembly and polyomino context-free grammars. Natural Computing, 14(2):293-302, 2015.
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Homotopy Measures for Representative Trajectories
An important task in trajectory analysis is defining a meaningful representative for a cluster of similar trajectories. Formally defining and computing such a representative r is a challenging problem. We propose and discuss two new definitions, both of which use only the geometry of the input trajectories. The definitions are based on the homotopy area as a measure of similarity between two curves, which is a minimum area swept by all possible deformations of one curve into the other. In the first definition we wish to minimize the maximum homotopy area between r and any input trajectory, whereas in the second definition we wish to minimize the sum of the homotopy areas between r and the input trajectories. For both definitions computing an optimal representative is NP-hard. However, for the case of minimizing the sum of the homotopy areas, an optimal representative can be found efficiently in a natural class of restricted inputs, namely, when the arrangement of trajectories forms a directed acyclic graph.
trajectory analysis
representative trajectory
homotopy area
27:1-27:17
Regular Paper
Erin
Chambers
Erin Chambers
Irina
Kostitsyna
Irina Kostitsyna
Maarten
Löffler
Maarten Löffler
Frank
Staals
Frank Staals
10.4230/LIPIcs.ESA.2016.27
Pankaj K. Agarwal, Mark de Berg, Jie Gao, Leonidas J. Guibas, and Sariel Har-Peled. Staying in the middle: Exact and approximate medians in R¹ and R² for moving points. In CCCG, pages 43-46, 2005.
Riddhipratim Basu, BhaswarB. Bhattacharya, and Tanmoy Talukdar. The projection median of a set of points in R^d. Discrete &Computational Geometry, 47(2):329-346, 2012.
Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo. Detecting commuting patterns by clustering subtrajectories. IJCGA, 21(03):253-282, 2011. URL: http://dx.doi.org/10.1142/S0218195911003652.
http://dx.doi.org/10.1142/S0218195911003652
Kevin Buchin, Maike Buchin, Marc Kreveld, Maarten Löffler, RodrigoI. Silveira, Carola Wenk, and Lionov Wiratma. Median trajectories. Algorithmica, 66(3):595-614, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9654-2.
http://dx.doi.org/10.1007/s00453-012-9654-2
Erin W. Chambers and David Letscher. On the height of a homotopy. In CCCG, pages 103-106, 2009.
Erin Wolf Chambers and Mikael Vejdemo-Johansson. Computing minimum area homologies. Computer Graphics Forum, 2014. URL: http://dx.doi.org/10.1111/cgf.12514.
http://dx.doi.org/10.1111/cgf.12514
Erin Wolf Chambers and Yusu Wang. Measuring similarity between curves on 2-manifolds via homotopy area. In Proc. 29th Ann. Symp. on CG, pages 425-434. ACM, 2013. URL: http://dx.doi.org/10.1145/2462356.2462375.
http://dx.doi.org/10.1145/2462356.2462375
Erin Wolf Chambers, Éric Colin de Verdière, Jeff Erickson, Sylvain Lazard, Francis Lazarus, and Shripad Thite. Homotopic fréchet distance between curves or, walking your dog in the woods in polynomial time. CG, 43(3):295-311, 2010. URL: http://dx.doi.org/10.1016/j.comgeo.2009.02.008.
http://dx.doi.org/10.1016/j.comgeo.2009.02.008
Timothy M Chan. On levels in arrangements of curves, iii: further improvements. In Proc. of the 24th annual symposium on Computational geometry, pages 85-93. ACM, 2008.
Stephane Durocher and David Kirkpatrick. The projection median of a set of points. CG, 42(5):364-375, 2009.
S. Gaffney, A. Robertson, P. Smyth, S. Camargo, and M. Ghil. Probabilistic clustering of extratropical cyclones using regression mixture models. Climate Dynamics, 29(4):423-440, 2007.
S. Gaffney and P. Smyth. Trajectory clustering with mixtures of regression models. In Proc. 5th ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining, pages 63-72, 1999.
Sariel Har-Peled, Amir Nayyeri, Mohammad Salavatipour, and Anastasios Sidiropoulos. How to walk your dog in the mountains with no magic leash. In Proc. 28th Ann. Symp. on CG, pages 121-130. ACM, 2012. URL: http://dx.doi.org/10.1145/2261250.2261269.
http://dx.doi.org/10.1145/2261250.2261269
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2001. URL: http://www.math.cornell.edu/~hatcher/.
http://www.math.cornell.edu/~hatcher/
Donald E. Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM Journal on Discrete Mathematics, 5(3):422-427, 1992. URL: http://dx.doi.org/10.1137/0405033.
http://dx.doi.org/10.1137/0405033
J.G. Lee, J. Han, and K.Y. Whang. Trajectory clustering: a partition-and-group framework. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 593-604, 2007.
David Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11(2):329-343, 1982. URL: http://dx.doi.org/10.1137/0211025.
http://dx.doi.org/10.1137/0211025
James R. Munkres. Topology. Prentice-Hall, 2nd edition, 2000.
M. Vlachos, D. Gunopulos, and G. Kollios. Discovering similar multidimensional trajectories. In Proc. 18th Int. Conf. Data Engin., pages 673-684, 2002.
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Optimal Reachability and a Space-Time Tradeoff for Distance Queries in Constant-Treewidth Graphs
We consider data-structures for answering reachability and distance queries on constant-treewidth graphs with n nodes, on the standard RAM computational model with wordsize W=Theta(log n). Our first contribution is a data-structure that after O(n) preprocessing time, allows (1) pair reachability queries in O(1) time; and (2) single-source reachability queries in O(n/log n) time. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. The data-structure uses at all times O(n) space. Our second contribution is a space-time tradeoff data-structure for distance queries. For any epsilon in [1/2,1], we provide a data-structure with polynomial preprocessing time that allows pair queries in O(n^{1-\epsilon} alpha(n)) time, where alpha is the inverse of the Ackermann function, and at all times uses O(n^epsilon) space. The input graph G is not considered in the space complexity.
Graph algorithms
Constant-treewidth graphs
Reachability queries
Distance queries
28:1-28:17
Regular Paper
Krishnendu
Chatterjee
Krishnendu Chatterjee
Rasmus
Rasmus Ibsen-Jensen
Rasmus Rasmus Ibsen-Jensen
Andreas
Pavlogiannis
Andreas Pavlogiannis
10.4230/LIPIcs.ESA.2016.28
Takuya Akiba, Yoichi Iwata, and Yuichi Yoshida. Fast exact shortest-path distance queries on large networks by pruned landmark labeling. In SIGMOD'13, SIGMOD'13, pages 349-360, 2013.
Takuya Akiba, Christian Sommer, and Ken-ichi Kawarabayashi. Shortest-Path Queries for Complex Networks: Exploiting Low Tree-width Outside the Core. In EDBT, pages 144-155, 2012.
Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees . Discrete Applied Mathematics, 23(1):11-24, 1989.
Reinhard Bauer, Tobias Columbus, Ignaz Rutter, and Dorothea Wagner. Search-space size in contraction hierarchies. In ICALP 13, pages 93-104, 2013.
R. Bellman. On a Routing Problem. Quarterly of Applied Mathematics, 16:87-90, 1958.
Michael A. Bender and Martín Farach-Colton. The LCA problem revisited. In LATIN 2000: Theoretical Informatics. Springer Berlin Heidelberg, 2000.
M.W Bern, E.L Lawler, and A.L Wong. Linear-time computation of optimal subgraphs of decomposable graphs. Journal of Algorithms, 8(2):216-235, 1987.
H. L. Bodlaender. Dynamic programming on graphs with bounded treewidth. In ICALP, volume LNCS 317, pages 105-118. Springer, 1988.
H. L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6), December 1996.
H. L. Bodlaender. Discovering treewidth. In SOFSEM'05, volume LNCS 3381, pages 1-16. Springer, 2005.
Hans L. Bodlaender. A tourist guide through treewidth. Acta Cybern., 11(1-2):1-21, 1993.
HansL. Bodlaender and Torben Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. SIAM Journal on Computing, 27:1725-1746, 1995.
Krishnendu Chatterjee, Amir Kafshdar Goharshady, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Algorithms for algebraic path properties in concurrent systems of constant treewidth components. In POPL, pages 733-747, 2016.
Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Prateesh Goyal, and Andreas Pavlogiannis. Faster algorithms for algebraic path properties in recursive state machines with constant treewidth. In POPL, 2015.
Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Faster algorithms for quantitative verification in constant treewidth graphs. In CAV, 2015.
Shiva Chaudhuri and Christos D. Zaroliagis. Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms. Algorithmica, 27:212-226, 1995.
Tobias Columbus. Search space size in contraction hierarchies. Master’s thesis, Karlsruhe Institute of Technology, 2012.
T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction To Algorithms. MIT Press, 2001.
Edsger. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269-271, 1959.
M. Elberfeld, A. Jakoby, and T. Tantau. Logspace versions of the theorems of Bodlaender and Courcelle. In FOCS, 2010.
Michael J. Fischer and Albert R. Meyer. Boolean Matrix Multiplication and Transitive Closure. In SWAT (FOCS), pages 129-131. IEEE Computer Society, 1971.
Robert W. Floyd. Algorithm 97: Shortest path. Communications of the ACM, 5(6):345, 1962.
Lester R. Ford. Network Flow Theory. Report P-923, The Rand Corporation, 1956.
Rudolf Halin. S-functions for graphs. Journal of Geometry, 8(1-2):171-186, 1976.
D. Harel and R. Tarjan. Fast Algorithms for Finding Nearest Common Ancestors. SIAM Journal on Computing, 13(2):338-355, 1984.
Peter E. Hart, Nils J. Nilsson, and Bertram Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. on Systems Science and Cybernetics, 4(2):100-107, 1968.
Donald B. Johnson. Efficient Algorithms for Shortest Paths in Sparse Networks. J. ACM, 24(1):1-13, January 1977.
Edward F. Moore. The shortest path through a maze. In Proceedings of the International Symposium on the Theory of Switching, and Annals of the Computation Laboratory of Harvard University, pages 285-292. Harvard University Press, 1959.
Leon R. Planken, Mathijs M. de Weerdt, and Roman P.J. van der Krogt. Computing all-pairs shortest paths by leveraging low treewidth. In ICAPS-11, pages 170-177. AAAI Press, 2011.
Neil Robertson and P.D Seymour. Graph minors. III. planar tree-width. Journal of Combinatorial Theory, Series B, 36(1):49-64, 1984.
B. Roy. Transitivité et connexité. C. R. Acad. Sci. Paris, 249:216-218, 1959.
Mikkel Thorup. All Structured Programs Have Small Tree Width and Good Register Allocation. Information and Computation, 142(2):159-181, 1998.
Stephen Warshall. A Theorem on Boolean Matrices. J. ACM, 9(1):11-12, January 1962.
Atsuko Yamaguchi, Kiyoko F. Aoki, and Hiroshi Mamitsuka. Graph complexity of chemical compounds in biological pathways. Genome Informatics, 14:376-377, 2003.
Yosuke Yano, Takuya Akiba, Yoichi Iwata, and Yuichi Yoshida. Fast and scalable reachability queries on graphs by pruned labeling with landmarks and paths. In CIKM'13, pages 1601-1606, 2013.
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An ILP-based Proof System for the Crossing Number Problem
Formally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations.
automatic formal proof
crossing number
integer linear programming
29:1-29:13
Regular Paper
Markus
Chimani
Markus Chimani
Tilo
Wiedera
Tilo Wiedera
10.4230/LIPIcs.ESA.2016.29
David Applegate, Robert E. Bixby, Vasek Chvátal, William J. Cook, Daniel G. Espinoza, Marcos Goycoolea, and Keld Helsgaun. Certification of an optimal TSP tour through 85, 900 cities. Oper. Res. Lett., 37(1):11-15, 2009. URL: http://dx.doi.org/10.1016/j.orl.2008.09.006.
http://dx.doi.org/10.1016/j.orl.2008.09.006
David Applegate, William J. Cook, Sanjeeb Dash, and Daniel G. Espinoza. Exact solutions to linear programming problems. Oper. Res. Lett., 35(6):693-699, 2007. URL: http://dx.doi.org/10.1016/j.orl.2006.12.010.
http://dx.doi.org/10.1016/j.orl.2006.12.010
Drago Bokal. On the crossing numbers of cartesian products with paths. Journal of Combinatorial Theory, Series B, 97(3):381-384, 2007. URL: http://dx.doi.org/10.1016/j.jctb.2006.06.003.
http://dx.doi.org/10.1016/j.jctb.2006.06.003
Christoph Buchheim, Markus Chimani, Dietmar Ebner, Carsten Gutwenger, Michael Jünger, Gunnar W. Klau, Petra Mutzel, and René Weiskircher. A branch-and-cut approach to the crossing number problem. Discrete Optimization, 5(2):373-388, 2008. URL: http://dx.doi.org/10.1016/j.disopt.2007.05.006.
http://dx.doi.org/10.1016/j.disopt.2007.05.006
Sergio Cabello. Hardness of approximation for crossing number. Discrete & Computational Geometry, 49(2):348-358, 2013. URL: http://dx.doi.org/10.1007/s00454-012-9440-6.
http://dx.doi.org/10.1007/s00454-012-9440-6
Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing, 42:1803-1829, 2013.
Markus Chimani. Computing Crossing Numbers. PhD thesis, TU Dortmund, 2008. URL: http://hdl.handle.net/2003/25955.
http://hdl.handle.net/2003/25955
Markus Chimani. Facets in the crossing number polytope. SIAM Journal on Discrete Mathematics, 25(1):95-111, 2011. URL: http://epubs.siam.org/sidma/resource/1/sjdmec/v25/i1/p95_s1, URL: http://dx.doi.org/10.1137/09076965X.
http://dx.doi.org/10.1137/09076965X
Markus Chimani and Carsten Gutwenger. Non-planar core reduction of graphs. Discrete Mathematics, 309(7):1838-1855, 2009. URL: http://dx.doi.org/10.1016/j.disc.2007.12.078.
http://dx.doi.org/10.1016/j.disc.2007.12.078
Markus Chimani, Carsten Gutwenger, Michael Jünger, Gunnar W. Klau, Karsten Klein, and Petra Mutzel. The open graph drawing framework (OGDF). In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization., pages 543-569. Chapman &Hall/CRC, 2013.
Markus Chimani, Carsten Gutwenger, and Petra Mutzel. Experiments on exact crossing minimization using column generation. ACM J. Experim. Alg., 14:4:3.4-4:3.18, 2010. URL: http://dx.doi.org/10.1145/1498698.1564504.
http://dx.doi.org/10.1145/1498698.1564504
Markus Chimani, Petra Mutzel, and Immanuel Bomze. A new approach to exact crossing minimization. In Proc. ESA, volume 5193 of LNCS, pages 284-296, 2008. URL: http://dx.doi.org/10.1007/978-3-540-87744-8_24.
http://dx.doi.org/10.1007/978-3-540-87744-8_24
Marcel Dhiflaoui, Stefan Funke, Carsten Kwappik, Kurt Mehlhorn, Michael Seel, Elmar Schömer, Ralph Schulte, and Dennis Weber. Certifying and repairing solutions to large LPs how good are LP-solvers? In Proc. Fourteenth SODA, pages 255-256. ACM/SIAM, 2003.
Giuseppe Di Battista, Ashim Garg, Giuseppe Liotta, Roberto Tamassia, Emanuele Tassinari, and Francesco Vargiu. An experimental comparison of four graph drawing algorithms. Computational Geometry, 7(5–6):303-325, 1997. 11th ACM Symposium on Computational Geometry. URL: http://dx.doi.org/10.1016/S0925-7721(96)00005-3.
http://dx.doi.org/10.1016/S0925-7721(96)00005-3
Guiseppe Di Battista, Ashim Garg, Guiseppe Liotta, Armando Parise, Roberto Tassinari, Emanuele Tassinari, Francesco Vargiu, and Luca Vismara. Drawing directed acyclic graphs: An experimental study. International Journal of Computational Geometry &Applications, 10(06):623-648, 2000. URL: http://dx.doi.org/10.1142/S0218195900000358.
http://dx.doi.org/10.1142/S0218195900000358
Geoffrey Exoo, Frank Harary, and Jerald Kabell. The crossing numbers of some generalized petersen graphs. Mathematica Scandinavica, 48(1):184-188, 1981.
M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983. URL: http://dx.doi.org/10.1137/0604033.
http://dx.doi.org/10.1137/0604033
Petr Hliněný. Crossing number is hard for cubic graphs. Journal of Combinatorial Theory. Series B, 96(4):455-471, 2006. URL: http://dx.doi.org/10.1016/j.jctb.2005.09.009.
http://dx.doi.org/10.1016/j.jctb.2005.09.009
Michael Jünger and Stefan Thienel. The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Softw., Pract. Exper., 30(11):1325-1352, 2000. URL: http://dx.doi.org/10.1002/1097-024X(200009)30:11<1325::AID-SPE342>3.0.CO;2-T.
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Ken-ichi Kawarabayashi and Buce Reed. Computing crossing number in linear time. In Proc. STOC, pages 382-390, 2007. URL: http://dx.doi.org/10.1145/1250790.1250848.
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Dan McQuillan, Shengjun Pan, and R. Bruce Richter. On the crossing number of K_13. Journal of Combinatorial Theory, Series B, 115:224-235, 2015. URL: http://dx.doi.org/10.1016/j.jctb.2015.06.002.
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Dan McQuillan and R. Bruce Richter. On the crossing numbers of certain generalized petersen graphs. Discrete Mathematics, 104(3):311-320, 1992. URL: http://dx.doi.org/10.1016/0012-365X(92)90453-M.
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Lars Noschinski, Christine Rizkallah, and Kurt Mehlhorn. Verification of certifying computations through autocorres and simpl. In Proc. NASA Formal Methods - 6th International Symposium, volume 8430 of LNCS, pages 46-61, 2014. URL: http://dx.doi.org/10.1007/978-3-319-06200-6_4.
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Shengjun Pan and R. Bruce Richter. The crossing number of K_11 is 100. Journal of Graph Theory, 56(2):128-134, 2007. URL: http://dx.doi.org/10.1002/jgt.v56:2.
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Strategic Contention Resolution with Limited Feedback
In this paper, we study contention resolution protocols from a game-theoretic perspective. We focus on acknowledgment-based protocols, where a user gets feedback from the channel only when she attempts transmission. In this case she will learn whether her transmission was successful or not. Users that do not transmit will not receive any feedback. We are interested in equilibrium protocols, where no player has an incentive to deviate.
The limited feedback makes the design of equilibrium protocols a hard task as best response policies usually have to be modeled as Partially Observable Markov Decision Processes, which are hard to analyze. Nevertheless, we show how to circumvent this for the case of two players and present an equilibrium protocol. For many players, we give impossibility results for a large class of acknowledgment-based protocols, namely age-based and backoff protocols with finite expected finishing time. Finally, we provide an age-based equilibrium protocol, which has infinite expected finishing time, but every player finishes in linear time with high probability.
contention resolution
acknowledgment-based protocols
game theory
30:1-30:16
Regular Paper
George
Christodoulou
George Christodoulou
Martin
Gairing
Martin Gairing
Sotiris
Nikoletseas
Sotiris Nikoletseas
Christoforos
Raptopoulos
Christoforos Raptopoulos
Paul
Spirakis
Paul Spirakis
10.4230/LIPIcs.ESA.2016.30
N. Abramson. The ALOHA system: Another alternative for computer communications. In Proceedings of the November 17-19, 1970, fall joint computer conference, pages 281-285. ACM New York, NY, USA, 1970.
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http://dx.doi.org/10.1016/j.comnet.2004.02.013
E. Altman, D. Barman, A. Benslimane, and R. El Azouzi. Slotted aloha with priorities and random power. In Proc. IEEE Infocom, 2005.
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http://dx.doi.org/10.1145/1288107.1288109
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Cell-Probe Lower Bounds for Bit Stream Computation
We revisit the complexity of online computation in the cell probe model. We consider a class of problems where we are first given a fixed pattern F of n symbols and then one symbol arrives at a time in a stream. After each symbol has arrived we must output some function of F and the n-length suffix of the arriving stream. Cell probe bounds of Omega(delta lg n/w) have previously been shown for both convolution and Hamming distance in this setting, where delta is the size of a symbol in bits and w in Omega(lg n) is the cell size in bits. However, when delta is a constant, as it is in many natural situations, the existing approaches no longer give us non-trivial bounds.
We introduce a lop-sided information transfer proof technique which enables us to prove meaningful lower bounds even for constant size input alphabets. Our new framework is capable of proving amortised cell probe lower bounds of Omega(lg^2 n/(w lg lg n)) time per arriving bit. We demonstrate this technique by showing a new lower bound for a problem known as pattern matching with address errors or the L_2-rearrangement distance problem. This gives the first non-trivial cell probe lower bound for any online problem on bit streams that still holds when the cell size is large.
Cell-probe lower bounds
algorithms
data streaming
31:1-31:15
Regular Paper
Raphaël
Clifford
Raphaël Clifford
Markus
Jalsenius
Markus Jalsenius
Benjamin
Sach
Benjamin Sach
10.4230/LIPIcs.ESA.2016.31
A. Amir, Y. Aumann, G. Benson, A. Levy, O. Lipsky, E. Porat, S. Skiena, and U. Vishne. Pattern matching with address errors: rearrangement distances. In SODA'06: Proc. 17superscriptth ACM-SIAM Symp. on Discrete Algorithms, pages 1221-1229. ACM Press, 2006.
A. Amir, Y. Aumann, G. Benson, A. Levy, O. Lipsky, E. Porat, S. Skiena, and U. Vishne. Pattern matching with address errors: Rearrangement distances. Journal of Computer System Sciences, 75(6):359-370, 2009.
R. Clifford and M. Jalsenius. Lower bounds for online integer multiplication and convolution in the cell-probe model. In ICALP'11: Proc. 38superscriptth International Colloquium on Automata, Languages and Programming, pages 593-604, 2011. URL: http://arxiv.org/abs/1101.0768.
http://arxiv.org/abs/1101.0768
R. Clifford, M. Jalsenius, and B. Sach. Tight cell-probe bounds for online hamming distance computation. In SODA'13: Proc. 24superscriptth ACM-SIAM Symp. on Discrete Algorithms, pages 664-674, 2013. URL: http://arxiv.org/abs/1207.1885.
http://arxiv.org/abs/1207.1885
R. Clifford, M. Jalsenius, and B. Sach. Cell-probe bounds for online edit distance and other pattern matching problems. In SODA'15: Proc. 26superscriptth ACM-SIAM Symp. on Discrete Algorithms, 2015. URL: http://arxiv.org/abs/1407.6559.
http://arxiv.org/abs/1407.6559
R. Clifford and B. Sach. Pattern matching in pseudo real-time. Journal of Discrete Algorithms, 9(1):67-81, 2011.
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R.Clifford, A. Grønlund, and K. Green Larsen. New unconditional hardness results for dynamic and online problems. In FOCS'15: Proc. 56superscriptth Annual Symp. Foundations of Computer Science, pages 1089-1107, 2015.
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Stochastic Streams: Sample Complexity vs. Space Complexity
We address the trade-off between the computational resources needed to process a large data set and the number of samples available from the data set. Specifically, we consider the following abstraction: we receive a potentially infinite stream of IID samples from some unknown distribution D, and are tasked with computing some function f(D). If the stream is observed for time t, how much memory, s, is required to estimate f(D)? We refer to t as the sample complexity and s as the space complexity. The main focus of this paper is investigating the trade-offs between the space and sample complexity. We study these trade-offs for several canonical problems studied in the data stream model: estimating the collision probability, i.e., the second moment of a distribution, deciding if a graph is connected, and approximating the dimension of an unknown subspace. Our results are based on techniques for simulating different classical sampling procedures in this model, emulating random walks given a sequence of IID samples, as well as leveraging a characterization between communication bounded protocols and statistical query algorithms.
data streams
sample complexity
frequency moments
graph connectivity
subspace approximation
32:1-32:15
Regular Paper
Michael
Crouch
Michael Crouch
Andrew
McGregor
Andrew McGregor
Gregory
Valiant
Gregory Valiant
David P.
Woodruff
David P. Woodruff
10.4230/LIPIcs.ESA.2016.32
Jayadev Acharya, Alon Orlitsky, Ananda Theertha Suresh, and Himanshu Tyagi. The complexity of estimating rényi entropy. CoRR, abs/1408.1000, 2014.
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. Journal of Computer and System Sciences, 58(1):137-147, 1999.
Alexandr Andoni, Andrew McGregor, Krzysztof Onak, and Rina Panigrahy. Better bounds for frequency moments in random-order streams. CoRR, abs/0808.2222, 2008.
Ziv Bar-Yossef, Ravi Kumar, and D. Sivakumar. Sampling algorithms: lower bounds and applications. In STOC, pages 266-275, 2001. URL: http://dx.doi.org/10.1145/380752.380810.
http://dx.doi.org/10.1145/380752.380810
Greg Barnes and Uriel Feige. Short Random Walks on Graphs. SIAM Journal on Discrete Mathematics, 9(1):19, 1996. URL: http://dx.doi.org/10.1137/S0895480194264988.
http://dx.doi.org/10.1137/S0895480194264988
Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. CoRR, abs/1505.02019, 2015. URL: http://arxiv.org/abs/1505.02019.
http://arxiv.org/abs/1505.02019
Amit Chakrabarti, Graham Cormode, and Andrew McGregor. Robust lower bounds for communication and stream computation. In STOC, pages 641-650, 2008. URL: http://dx.doi.org/10.1145/1374376.1374470.
http://dx.doi.org/10.1145/1374376.1374470
Amit Chakrabarti, T. S. Jayram, and Mihai Patrascu. Tight lower bounds for selection in randomly ordered streams. In SODA, pages 720-729, 2008. URL: http://dx.doi.org/10.1145/1347082.1347161.
http://dx.doi.org/10.1145/1347082.1347161
Steve Chien, Katrina Ligett, and Andrew McGregor. Space-efficient estimation of robust statistics and distribution testing. In ICS, pages 251-265, 2010.
Kenneth L. Clarkson and David P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 205-214, 2009.
Thomas M. Cover, Michael A. Freedman, and Martin E. Hellman. Optimal finite memory learning algorithms for the finite sample problem. Information and Control, 30(1):49-85, 1976.
Klim Efremenko and Omer Reingold. How Well Do Random Walks Parallelize? In APPROX-RANDOM, volume 5687, pages 476-489, Berlin, Heidelberg, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03685-9.
http://dx.doi.org/10.1007/978-3-642-03685-9
Uriel Feige. A fast randomized logspace algorithm for graph connectivity. Theor. Comput. Sci., 169(2):147-160, 1996. URL: http://dx.doi.org/10.1016/S0304-3975(96)00118-1.
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Uriel Feige. A Spectrum of Time-Space Trade-offs for Undirected s-t Connectivity. Journal of Computer and System Sciences, 54(2):305-316, April 1997. URL: http://dx.doi.org/10.1006/jcss.1997.1471.
http://dx.doi.org/10.1006/jcss.1997.1471
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Michael Greenwald and Sanjeev Khanna. Efficient online computation of quantile summaries. In ACM International Conference on Management of Data, pages 58-66, 2001. URL: http://dx.doi.org/10.1145/375663.375670.
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Sudipto Guha and Zhiyi Huang. Revisiting the direct sum theorem and space lower bounds in random order streams. In ICALP, pages 513-524, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02927-1_43.
http://dx.doi.org/10.1007/978-3-642-02927-1_43
Sudipto Guha and Andrew McGregor. Space-efficient sampling. In AISTATS, pages 169-176, 2007.
Sudipto Guha and Andrew McGregor. Stream order and order statistics: Quantile estimation in random-order streams. SIAM Journal on Computing, 38(5):2044-2059, 2009. URL: http://dx.doi.org/10.1137/07069328X.
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Martin E Hellman and Thomas M Cover. Learning with finite memory. The Annals of Mathematical Statistics, pages 765-782, 1970.
Daniel M. Kane, Jelani Nelson, and David P. Woodruff. An optimal algorithm for the distinct elements problem. In PODS, pages 41-52, 2010. URL: http://dx.doi.org/10.1145/1807085.1807094.
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Yi Li, Huy L. Nguyen, and David P. Woodruff. On sketching matrix norms and the top singular vector. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1562-1581, 2014.
Andrew McGregor, A. Pavan, Srikanta Tirthapura, and David P. Woodruff. Space-efficient estimation of statistics over sub-sampled streams. In PODS, pages 273-282, 2012. URL: http://dx.doi.org/10.1145/2213556.2213594.
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Andrew McGregor and Paul Valiant. The shifting sands algorithm. In ACM-SIAM Symposium on Discrete Algorithms, pages 453-458, 2012.
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S. Muthukrishnan. Stochastic data streams. In MFCS, page 55, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03816-7_5.
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Piyush Rai, Hal Daumé III, and Suresh Venkatasubramanian. Streamed learning: One-pass SVMs. In IJCAI, pages 1211-1216, 2009.
Ran Raz. Fast learning requires good memory: A time-space lower bound for parity learning. CoRR, abs/1602.05161, 2016.
Omer Reingold. Undirected connectivity in log-space. Journal of the ACM, 55(4):1-24, September 2008. URL: http://dx.doi.org/10.1145/1391289.1391291.
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Atri Rudra and Steve Uurtamo. Data stream algorithms for codeword testing. In Automata, Languages and Programming, pages 629-640. Springer, 2010.
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Ohad Shamir. Fundamental limits of online and distributed algorithms for statistical learning and estimation. In Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pages 163-171, 2014.
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David P. Woodruff. The average-case complexity of counting distinct elements. In ICDT, pages 284-295, 2009. URL: http://dx.doi.org/10.1145/1514894.1514928.
http://dx.doi.org/10.1145/1514894.1514928
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Counting Matchings with k Unmatched Vertices in Planar Graphs
We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [Kasteleyn 1961], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [Jerrum 1987].
To interpolate between matchings and perfect matchings, we study the parameterized problem of counting matchings with k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that become planar after removing k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs [Curtican and Xia 2015], we obtain:
- Counting matchings with k unmatched vertices in planar graphs is #W[1]-hard.
- In contrast, given a plane graph G with s distinguished faces, there is an O(2^s n^3) time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an f(k,s)n^O(1) time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.
counting complexity
parameterized complexity
matchings
planar graphs
33:1-33:17
Regular Paper
Radu
Curticapean
Radu Curticapean
10.4230/LIPIcs.ESA.2016.33
Manindra Agrawal. Determinant versus permanent. In Proceedings of the 25th International Congress of Mathematicians, ICM 2006, volume 3, pages 985-997, 2006.
Markus Bläser and Radu Curticapean. Weighted counting of k-matchings is #W[1]-hard. In IPEC, pages 171-181, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33293-7_17.
http://dx.doi.org/10.1007/978-3-642-33293-7_17
P. Bürgisser. Completeness and Reduction in Algebraic Complexity Theory. Number 7 in Algorithms and Computation in Mathematics. Springer Verlag, 2000. 168 + xii pp.
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Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, and Ge Xia. Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput., 201(2):216-231, 2005.
Radu Curticapean. Counting matchings of size k is #W[1]-hard. In ICALP 2013, pages 352-363, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_30.
http://dx.doi.org/10.1007/978-3-642-39206-1_30
Radu Curticapean. Counting perfect matchings in graphs that exclude a single-crossing minor. CoRR, abs/1406.4056, 2014.
Radu Curticapean. Block interpolation: A framework for tight exponential-time counting complexity. In ICALP 2015, pages 380-392, 2015.
Radu Curticapean. Parity separation: A scientifically proven method for permanent weight loss. CoRR, abs/1511.07480, 2015.
Radu Curticapean. The simple, little and slow things count: on parameterized counting complexity. PhD thesis, Saarland University, 2015.
Radu Curticapean and Dániel Marx. Complexity of counting subgraphs: Only the boundedness of the vertex-cover number counts. In FOCS 2014, pages 130-139, 2014.
Radu Curticapean and Mingji Xia. Parameterizing the permanent: Genus, apices, minors, evaluation mod 2^k. In FOCS 2015, pages 994-1009, 2015.
Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential time complexity of the permanent and the Tutte polynomial. ACM Transactions on Algorithms, 10(4):21, 2014.
Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. Approximation algorithms via structural results for apex-minor-free graphs. In ICALP 2009, pages 316-327, 2009.
Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM Journal on Computing, 33(4):892-922, 2004.
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006.
Markus Frick. Generalized model-checking over locally tree-decomposable classes. Theory Comput. Syst., 37(1):157-191, 2004.
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Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. System Sci., 63(4):512-530, 2001.
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On Interference Among Moving Sensors and Related Problems
We show that for any set of n moving points in R^d and any parameter 2<=k<n, one can select a fixed non-empty subset of the points of size O(k log k), such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains O(n/k) points per cell). We also show that the bound O(k log k) is near optimal even for the one dimensional case in which points move linearly in time. As an application, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time, their interference is O( (n log n)^0.5). This is optimal up to an O((log n)^0.5) factor.
Range spaces
Voronoi diagrams
moving points
facility location
interference minimization
34:1-34:11
Regular Paper
Jean-Lou
De Carufel
Jean-Lou De Carufel
Matthew J.
Katz
Matthew J. Katz
Matias
Korman
Matias Korman
André
van Renssen
André van Renssen
Marcel
Roeloffzen
Marcel Roeloffzen
Shakhar
Smorodinsky
Shakhar Smorodinsky
10.4230/LIPIcs.ESA.2016.34
N. Alon. A non-linear lower bound for planar epsilon-nets. Discrete & Computational Geometry, 47(2):235-244, 2012. URL: http://dx.doi.org/10.1007/s00454-010-9323-7.
http://dx.doi.org/10.1007/s00454-010-9323-7
B. Aronov, E. Ezra, and M. Sharir. Small-size epsilon-nets for axis-parallel rectangles and boxes. SIAM Journal on Computing, 39(7):3248-3282, 2010.
K. Böröczky. Packing of spheres in spaces of constant curvature. Acta Mathematica Academiae Scientiarum Hungarica, 32(3-4):243-261, 1978. URL: http://dx.doi.org/10.1007/BF01902361.
http://dx.doi.org/10.1007/BF01902361
Y. Brise, K. Buchin, D. Eversmann, M. Hoffmann, and W. Mulzer. Interference minimization in asymmetric sensor networks. In ALGOSENSORS 2014, pages 136-151, 2014. URL: http://dx.doi.org/10.1007/978-3-662-46018-4_9.
http://dx.doi.org/10.1007/978-3-662-46018-4_9
J.-L. De Carufel, M. Katz, M. Korman, A. van Renssen, M. Roeloffzen, and S. Smorodinsky. On kinetic range spaces and their applications. CoRR, abs/1507.02130, 2015. URL: http://arxiv.org/abs/1507.02130.
http://arxiv.org/abs/1507.02130
M.M. Halldórsson and T. Tokuyama. Minimizing interference of a wireless ad-hoc network in a plane. Theoretical Computer Science, 402(1):29-42, 2008. URL: http://dx.doi.org/10.1016/j.tcs.2008.03.003.
http://dx.doi.org/10.1016/j.tcs.2008.03.003
D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127-151, 1987.
J. Komlós, J. Pach, and G.J. Woeginger. Almost tight bounds for epsilon-nets. Discrete & Computational Geometry, 7:163-173, 1992.
M. Korman. Minimizing interference in ad-hoc networks with bounded communication radius. Information Processing Letters, 112(19):748-752, 2012. URL: http://dx.doi.org/10.1016/j.ipl.2012.06.021.
http://dx.doi.org/10.1016/j.ipl.2012.06.021
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J. Pach and G. Tardos. Tight lower bounds for the size of epsilon-nets. In Symposium on Computational Geometry, pages 458-463, 2011.
M. Sharir and P. K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995.
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P. von Rickenbach, R. Wattenhofer, and A. Zollinger. Algorithmic models of interference in wireless ad hoc and sensor networks. IEEE/ACM transactions on sensor networks, 17(1):172-185, 2009. URL: http://dx.doi.org/10.1109/TNET.2008.926506.
http://dx.doi.org/10.1109/TNET.2008.926506
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SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse
In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice.
Rips filtration
Homology groups
Persistence
Topological data analysis
35:1-35:16
Regular Paper
Tamal K.
Dey
Tamal K. Dey
Dayu
Shi
Dayu Shi
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.ESA.2016.35
H. Adams and G. Carlsson. On the nonlinear statistics of range image patches. SIAM J. Img. Sci., 2(1):110-117, 2009. URL: http://dx.doi.org/10.1137/070711669.
http://dx.doi.org/10.1137/070711669
U. Bauer, M. Kerber, and J. Reininghaus. Topological Methods in Data Analysis and Visualization III: Theory, Algorithms, and Applications, chapter Clear and Compress: Computing Persistent Homology in Chunks, pages 103-117. Springer International Publishing, Cham, 2014. URL: http://dx.doi.org/10.1007/978-3-319-04099-8_7.
http://dx.doi.org/10.1007/978-3-319-04099-8_7
U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner. Mathematical Software - ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings, chapter PHAT - Persistent Homology Algorithms Toolbox, pages 137-143. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014. Project URL: https://bitbucket.org/phat-code/phat.
https://bitbucket.org/phat-code/phat
J.-D. Boissonnat, T. K. Dey, and C. Maria. Algorithms - ESA 2013: 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, chapter The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology, pages 695-706. Springer, Berlin, Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_59.
http://dx.doi.org/10.1007/978-3-642-40450-4_59
J.-D. Boissonnat and C. Maria. Algorithms - ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10-12, 2012. Proceedings, chapter The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes, pages 731-742. Springer, 2012.
M. Buchet, F. Chazal, S. Y. Oudot, and D. R. Sheehy. Efficient and robust persistent homology for measures. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 168-180, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.13.
http://dx.doi.org/10.1137/1.9781611973730.13
D. Burghelea and T. K. Dey. Topological persistence for circle-valued maps. Discrete Comput. Geom., 50:69-98, 2013.
G. Carlsson. Topology and data. Bull. Amer. Math. Soc., 46:255-308, 2009.
G. Carlsson and V. de Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367-405, 2010.
G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009. URL: http://dx.doi.org/10.1007/s00454-009-9176-0.
http://dx.doi.org/10.1007/s00454-009-9176-0
N. J. Cavanna, M. Jahanseir, and D. R. Sheehy. A geometric perspective on sparse filtrations. In Canadian Conf. Comput. Geom. (CCCG), 2015. URL: http://dblp.uni-trier.de/db/conf/cccg/cccg2015.html#CavannaJS15.
http://dblp.uni-trier.de/db/conf/cccg/cccg2015.html#CavannaJS15
F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, SCG'09, pages 237-246, New York, NY, USA, 2009. ACM. URL: http://dx.doi.org/10.1145/1542362.1542407.
http://dx.doi.org/10.1145/1542362.1542407
F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, and S. Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. In Proc. of SGP, 2009.
F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. CoRR, abs/1207.3674, 2012.
F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba. Persistence-based clustering in Riemannian manifolds. In Proc. 27th Annu. ACM Sympos. Comput. Geom., pages 97-106, 2011.
C. Chen and M. Kerber. An output-sensitive algorithm for persistent homology. Comput. Geom. Theory Appl., 46(4):435-447, May 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2012.02.010.
http://dx.doi.org/10.1016/j.comgeo.2012.02.010
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics, 9(1):79-103, 2009.
D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the twenty-second annual symposium on Computational geometry, pages 119-126. ACM, 2006.
V. de Silva, D. Morozov, and M. Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete Comput. Geom., 45(4):737-759, 2011.
T. K. Dey, F. Fan, and Y. Wang. Computing topological persistence for simplicial maps. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 345-354. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582165.
http://dx.doi.org/10.1145/2582112.2582165
T. K. Dey, K. Li, C. Luo, P. Ranjan, I. Safa, and Y. Wang. Persistent heat signature for pose-oblivious matching of incomplete models. Comput. Graph. Forum. (special issue from Sympos. Geom. Process.), 29(5):1545-1554, 2010.
Dmitriy Morozov. Dionysus Software. http://mrzv.org/software/dionysus/, 2012.
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H. Edelsbrunner and J. Harer. Computational Topology - an Introduction. American Mathematical Society, 2010.
H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.
P. Frosini. A distance for similarity classes of submanifolds of a euclidean space. Bulletin of the Australian Mathematical Society, 42(3):407-416, 1990.
W. Harvey, I.-H. Park, O. Rübel, V. Pascucci, P.-T. Bremer, C. Li, and Y. Wang. A collaborative visual analytics suite for protein folding research. Journal of Molecular Graphics and Modelling, 53:59-71, 2014. URL: http://dx.doi.org/10.1016/j.jmgm.2014.06.003.
http://dx.doi.org/10.1016/j.jmgm.2014.06.003
M. Lichman. UCI machine learning repository, 2013. Project URL: http://archive.ics.uci.edu/ml.
http://archive.ics.uci.edu/ml
R. C. B. Madeo, S. M. Peres, and C. A. de M. Lima. Gesture phase segmentation using support vector machines. Expert Systems with Applications, 56:100-115, 2016. URL: http://dx.doi.org/10.1016/j.eswa.2016.02.021.
http://dx.doi.org/10.1016/j.eswa.2016.02.021
R. C. B. Madeo, P. K. Wagner, and S. M. Peres. Gesture Phase Segmentation Data Set, 2014. Project URL: http://archive.ics.uci.edu/ml/datasets/Gesture+Phase+Segmentation.
http://archive.ics.uci.edu/ml/datasets/Gesture+Phase+Segmentation
D. M. Mount and S. Arya. ANN: A Library for Approximate Nearest Neighbor Searching, 2010. Project URL: https://www.cs.umd.edu/~mount/ANN/.
https://www.cs.umd.edu/~mount/ANN/
James R. Munkres. Elements of Algebraic Topology. Addison-Wesley, 1993.
J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt. A stable multi-scale kernel for topological machine learning. In Proc. IEEE Conf. Comp. Vision &Pat. Rec. (CVPR), pages 4741-4748, 2015.
V. Robins. Towards computing homology from finite approximations. Topology Proceedings, 24(1):503-532, 1999.
D. R. Sheehy. Linear-size approximations to the vietoris-rips filtration. In Proceedings of the Twenty-eighth Annual Symposium on Computational Geometry, SoCG'12, pages 239-248. ACM, 2012. URL: http://dx.doi.org/10.1145/2261250.2261286.
http://dx.doi.org/10.1145/2261250.2261286
Simpers Software, 2015. Project URL: http://web.cse.ohio-state.edu/~tamaldey/SimPers/Simpers.html.
http://web.cse.ohio-state.edu/~tamaldey/SimPers/Simpers.html
G. Singh, F. Mémoli, T. Ishkhanov, G. Sapiro, G. Carlsson, and D. L Ringach. Topological analysis of population activity in visual cortex. Journal of vision, 8(8):11, 2008.
The GUDHI Project. GUDHI user and reference manual, 2015. Project URL: http://gudhi.gforge.inria.fr/doc/latest/.
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A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33(2):249-274, 2005.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Exponential Time Paradigms Through the Polynomial Time Lens
We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems.
As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms', and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs.
As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model.
exponential time paradigms
branching
dynamic programming
lower bounds
36:1-36:14
Regular Paper
Andrew
Drucker
Andrew Drucker
Jesper
Nederlof
Jesper Nederlof
Rahul
Santhanam
Rahul Santhanam
10.4230/LIPIcs.ESA.2016.36
Michael Alekhnovich, Allan Borodin, Joshua Buresh-Oppenheim, Russell Impagliazzo, Avner Magen, and Toniann Pitassi. Toward a model for backtracking and dynamic programming. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, pages 308-322. IEEE Computer Society, 2005. URL: http://dx.doi.org/10.1109/CCC.2005.32.
http://dx.doi.org/10.1109/CCC.2005.32
Eric Allender, Shiteng Chen, Tiancheng Lou, Periklis A. Papakonstantinou, and Bangsheng Tang. Width-parametrized SAT: time-space tradeoffs. Theory of Computing, 10:297-339, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a012.
http://dx.doi.org/10.4086/toc.2014.v010a012
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
http://dx.doi.org/10.1016/j.jcss.2009.04.001
Liming Cai and Jianer Chen. On the amount of nondeterminism and the power of verifying. SIAM Journal on Computing, 26(3):733-750, 1997. URL: http://dx.doi.org/10.1137/S0097539793258295.
http://dx.doi.org/10.1137/S0097539793258295
Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli. Extended formulations in combinatorial optimization. 4OR, 8(1):1-48, 2010. URL: http://dx.doi.org/10.1007/s10288-010-0122-z.
http://dx.doi.org/10.1007/s10288-010-0122-z
Marek Cygan, Fedor Fomin, Bart M.P. Jansen, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, and Saket Saurabh Michal Pilipczuk. Open problems for fpt school 2014.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Evgeny Dantsin and Edward A. Hirsch. Satisfiability certificates verifiable in subexponential time. In Karem A. Sakallah and Laurent Simon, editors, Theory and Applications of Satisfiability Testing - SAT 2011 - 14th International Conference, SAT 2011, Ann Arbor, MI, USA, June 19-22, 2011. Proceedings, volume 6695 of Lecture Notes in Computer Science, pages 19-32. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-21581-0_4.
http://dx.doi.org/10.1007/978-3-642-21581-0_4
Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: http://dx.doi.org/10.1145/2629620.
http://dx.doi.org/10.1145/2629620
Andrew Drucker. Nondeterministic direct product reductions and the success probability of SAT solvers. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 736-745. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.84.
http://dx.doi.org/10.1109/FOCS.2013.84
Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: http://dx.doi.org/10.1137/130927115.
http://dx.doi.org/10.1137/130927115
David Eppstein. Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms. ACM Trans. Algorithms, 2(4):492-509, October 2006. URL: http://dx.doi.org/10.1145/1198513.1198515.
http://dx.doi.org/10.1145/1198513.1198515
Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
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Oded Goldreich. Computational complexity - a conceptual perspective. Cambridge University Press, 2008.
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Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
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Dániel Marx. What’s next? future directions in parameterized complexity. In Hans L. Bodlaender, Rod Downey, Fedor V. Fomin, and Dániel Marx, editors, The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 469-496. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_20.
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Jesper Nederlof. Space and Time Efficient Structural Improvements of Dynamic Programming Algorithms. PhD thesis, University of Bergen, 2011. URL: http://www.win.tue.nl/~jnederlo/PhDThesis.pdf.
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Ramamohan Paturi and Pavel Pudlák. On the complexity of circuit satisfiability. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 241-250. ACM, 2010. URL: http://dx.doi.org/10.1145/1806689.1806724.
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Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, volume 47 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.57.
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Rahul Santhanam. On separators, segregators and time versus space. In Computational Complexity, 16th Annual IEEE Conference on, 2001., pages 286-294, 2001. URL: http://dx.doi.org/10.1109/CCC.2001.933895.
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Ryan Williams. Inductive time-space lower bounds for sat and related problems. computational complexity, 15(4):433-470, 2006.
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On the Power of Advice and Randomization for Online Bipartite Matching
While randomized online algorithms have access to a sequence of uniform random bits, deterministic online algorithms with advice have access to a sequence of advice bits, i.e., bits that are set by an all-powerful oracle prior to the processing of the request sequence. Advice bits are at least as helpful as random bits, but how helpful are they? In this work, we investigate the power of advice bits and random bits for online maximum bipartite matching (MBM).
The well-known Karp-Vazirani-Vazirani algorithm [Karp, Vazirani and Vazirani 90] is an optimal randomized (1-1/e)-competitive algorithm for MBM that requires access to Theta(n log n) uniform random bits. We show that Omega(log(1/epsilon) n) advice bits are necessary and O(1/epsilon^5 n) sufficient in order to obtain a (1-epsilon)-competitive deterministic advice algorithm. Furthermore, for a large natural class of deterministic advice algorithms, we prove that Omega(log log log n) advice bits are required in order to improve on the 1/2-competitiveness of the best deterministic online algorithm, while it is known that O(log n) bits are sufficient [Böckenhauer, Komm, Královic and Královic 2011].
Last, we give a randomized online algorithm that uses cn random bits, for integers c >= 1, and a competitive ratio that approaches 1-1/e very quickly as c is increasing. For example if c = 10, then the difference between 1-1/e and the achieved competitive ratio is less than 0.0002.
On-line algorithms
Bipartite matching
Randomization
37:1-37:16
Regular Paper
Christoph
Dürr
Christoph Dürr
Christian
Konrad
Christian Konrad
Marc
Renault
Marc Renault
10.4230/LIPIcs.ESA.2016.37
Anna Adamaszek, Marc P. Renault, Adi Rosén, and Rob van Stee. Reordering buffer management with advice. In 11th International Workshop on Approximation and Online Algorithms (WAOA), pages 132-143, September 2013.
Spyros Angelopoulos, Christoph Dürr, Shahin Kamali, Marc Renault, and Adi Rosén. Online bin packing with advice of small size. In Frank Dehne, Jörg-Rüdiger Sack, and Ulrike Stege, editors, Proceedings of the 14th International Symposium on Algorithms and Data Structures (WADS), pages 40-53. Springer International Publishing, August 2015.
Bahman Bahmani and Michael Kapralov. Improved bounds for online stochastic matching. In Proceedings of the 18th Annual European Conference on Algorithms (ESA), pages 170-181. Springer-Verlag, 2010.
Maria Paola Bianchi, Hans-Joachim Böckenhauer, Tatjana Brülisauer, Dennis Komm, and Beatrice Palano. Online minimum spanning tree with advice. In Proceedings of the 42nd International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM), pages 195-207, January 2016. URL: http://dx.doi.org/10.1007/978-3-662-49192-8_16.
http://dx.doi.org/10.1007/978-3-662-49192-8_16
Benjamin Birnbaum and Claire Mathieu. On-line bipartite matching made simple. SIGACT News, 39(1):80-87, March 2008. URL: http://dx.doi.org/10.1145/1360443.1360462.
http://dx.doi.org/10.1145/1360443.1360462
Hans-Joachim Böckenhauer, Juraj Hromkovic, Dennis Komm, Sacha Krug, Jasmin Smula, and Andreas Sprock. The string guessing problem as a method to prove lower bounds on the advice complexity. Theor. Comput. Sci., 554:95-108, 2014.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královič, Richard Královič, and Tobias Mömke. On the advice complexity of online problems. In Yingfei Dong, Ding-Zhu Du, and Oscar Ibarra, editors, Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC), pages 331-340. Springer Berlin Heidelberg, December 2009.
Hans-Joachim Böckenhauer, Dennis Komm, Richard Královic, and Peter Rossmanith. The online knapsack problem: Advice and randomization. Theor. Comput. Sci., 527:61-72, 2014.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královič, and Richard Královič. On the advice complexity of the k-server problem. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP), volume 6755 of Lecture Notes in Computer Science, pages 207-218. Springer Berlin Heidelberg, July 2011. URL: http://dx.doi.org/10.1007/978-3-642-22006-7_18.
http://dx.doi.org/10.1007/978-3-642-22006-7_18
Joan Boyar, Shahin Kamali, Kim S. Larsen, and Alejandro López-Ortiz. On the list update problem with advice. In Proceedings of the 8th International Conference on Language and Automata Theory and Applications (LATA), pages 210-221, March 2014. URL: http://dx.doi.org/10.1007/978-3-319-04921-2_17.
http://dx.doi.org/10.1007/978-3-319-04921-2_17
Joan Boyar, Shahin Kamali, Kim S. Larsen, and Alejandro López-Ortiz. Online bin packing with advice. Algorithmica, 74(1):507-527, 2016. URL: http://dx.doi.org/10.1007/s00453-014-9955-8.
http://dx.doi.org/10.1007/s00453-014-9955-8
Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized primal-dual analysis of RANKING for online bipartite matching. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 101-107, January 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.7.
http://dx.doi.org/10.1137/1.9781611973105.7
Stefan Dobrev, Rastislav Královič, and Dana Pardubská. How much information about the future is needed? In Proceedings of the 34th conference on Current trends in theory and practice of computer science (SOFSEM), pages 247-258, Berlin, Heidelberg, 2008. Springer-Verlag.
Sebastian Eggert, Lasse Kliemann, Peter Munstermann, and Anand Srivastav. Bipartite matching in the semi-streaming model. Algorithmica, 63(1):490-508, 2011.
Yuval Emek, Pierre Fraigniaud, Amos Korman, and Adi Rosén. Online computation with advice. Theor. Comput. Sci., 412(24):2642-2656, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.08.007.
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http://dx.doi.org/10.1007/978-3-642-32512-0_20
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Jesper W. Mikkelsen. Randomization can be as helpful as a glimpse of the future in online computation. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), July 2016. URL: http://arxiv.org/abs/1511.05886.
http://arxiv.org/abs/1511.05886
Shuichi Miyazaki. On the advice complexity of online bipartite matching and online stable marriage. Inf. Process. Lett., 114(12):714-717, December 2014. URL: http://dx.doi.org/10.1016/j.ipl.2014.06.013.
http://dx.doi.org/10.1016/j.ipl.2014.06.013
Marc P. Renault and Adi Rosén. On online algorithms with advice for the k-server problem. Theory Comput. Syst., 56(1):3-21, 2015. URL: http://dx.doi.org/10.1007/s00224-012-9434-z.
http://dx.doi.org/10.1007/s00224-012-9434-z
Marc P. Renault, Adi Rosén, and Rob van Stee. Online algorithms with advice for bin packing and scheduling problems. Theor. Comput. Sci., 600:155-170, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.07.050.
http://dx.doi.org/10.1016/j.tcs.2015.07.050
Daniel D. Sleator and Robert E. Tarjan. Amortized efficiency of list update and paging rules. Commun. ACM, 28(2):202-208, February 1985. URL: http://dx.doi.org/10.1145/2786.2793.
http://dx.doi.org/10.1145/2786.2793
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BlockQuicksort: Avoiding Branch Mispredictions in Quicksort
Since the work of Kaligosi and Sanders (2006), it is well-known that Quicksort - which is commonly considered as one of the fastest in-place sorting algorithms - suffers in an essential way from branch mispredictions. We present a novel approach to address this problem by partially decoupling control from data flow: in order to perform the partitioning, we split the input in blocks of constant size (we propose 128 data elements); then, all elements in one block are compared with the pivot and the outcomes of the comparisons are stored in a buffer. In a second pass, the respective elements are rearranged. By doing so, we avoid conditional branches based on outcomes of comparisons at all (except for the final Insertionsort). Moreover, we prove that for a static branch predictor the average total number of branch mispredictions is at most epsilon n log n + O(n) for some small epsilon depending on the block size when sorting n elements.
Our experimental results are promising: when sorting random integer data, we achieve an increase in speed (number of elements sorted per second) of more than 80% over the GCC implementation of C++ std::sort. Also for many other types of data and non-random inputs, there is still a significant speedup over std::sort. Only in few special cases like sorted or almost sorted inputs, std::sort can beat our implementation. Moreover, even on random input permutations, our implementation is even slightly faster than an implementation of the highly tuned Super Scalar Sample Sort, which uses a linear amount of additional space.
in-place sorting
Quicksort
branch mispredictions
lean programs
38:1-38:16
Regular Paper
Stefan
Edelkamp
Stefan Edelkamp
Armin
Weiss
Armin Weiss
10.4230/LIPIcs.ESA.2016.38
D. Abhyankar and M. Ingle. Engineering of a quicksort partitioning algorithm. Journal of Global Research in Computer Science, 2(2):17-23, 2011.
ARMv8 Instruction Set Overview, 2011. Document number: PRD03-GENC-010197 15.0.
Martin Aumüller and Martin Dietzfelbinger. Optimal partitioning for dual pivot quicksort - (extended abstract). In ICALP, pages 33-44, 2013.
Martin Aumüller, Martin Dietzfelbinger, and Pascal Klaue. How good is multi-pivot quicksort? CoRR, abs/1510.04676, 2015.
Paul Biggar, Nicholas Nash, Kevin Williams, and David Gregg. An experimental study of sorting and branch prediction. J. Exp. Algorithmics, 12:1.8:1-39, 2008.
Gerth Stølting Brodal, Rolf Fagerberg, and Kristoffer Vinther. Engineering a cache-oblivious sorting algorithm. J. Exp. Algorithmics, 12:2.2:1-23, 2008.
Gerth Stølting Brodal and Gabriel Moruz. Tradeoffs between branch mispredictions and comparisons for sorting algorithms. In WADS, volume 3608 of LNCS, pages 385-395. Springer, 2005.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. The MIT Press, 3nd edition, 2009.
Stefan Edelkamp and Armin Weiß. Blockquicksort: How branch mispredictions don't affect quicksort. CoRR, abs/1604.06697, 2016.
Amr Elmasry and Jyrki Katajainen. Lean programs, branch mispredictions, and sorting. In FUN, volume 7288 of LNCS, pages 119-130. Springer, 2012.
Amr Elmasry, Jyrki Katajainen, and Max Stenmark. Branch mispredictions don't affect mergesort. In SEA, pages 160-171, 2012.
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Kanela Kaligosi and Peter Sanders. How branch mispredictions affect quicksort. In ESA, pages 780-791, 2006.
Jyrki Katajainen. Sorting programs executing fewer branches. CPH STL Report 2263887503, Department of Computer Science, University of Copenhagen, 2014.
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Shrinu Kushagra, Alejandro López-Ortiz, Aurick Qiao, and J. Ian Munro. Multi-pivot quicksort: Theory and experiments. In ALENEX, pages 47-60, 2014.
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Conrado Martínez, Markus E. Nebel, and Sebastian Wild. Analysis of branch misses in quicksort. In Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2015, San Diego, CA, USA, January 4, 2015, pages 114-128, 2015.
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Sebastian Wild, Markus E. Nebel, and Ralph Neininger. Average case and distributional analysis of dual-pivot quicksort. ACM Transactions on Algorithms, 11(3):22:1-42, 2015.
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http://codeblab.com/wp-content/uploads/2009/09/DualPivotQuicksort.pdf
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Counting Linear Extensions: Parameterizations by Treewidth
We consider the #P-complete problem of counting the number of linear extensions of a poset (#LE); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of #LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that #LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that #LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.
Partially ordered sets
Linear extensions
Parameterized Complexity
Structural parameters
Treewidth
39:1-39:18
Regular Paper
Eduard
Eiben
Eduard Eiben
Robert
Ganian
Robert Ganian
Kustaa
Kangas
Kustaa Kangas
Sebastian
Ordyniak
Sebastian Ordyniak
10.4230/LIPIcs.ESA.2016.39
Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. J. Algorithms, 12(2):308-340, 1991.
Mike D Atkinson. On computing the number of linear extensions of a tree. Order, 7(1):23-25, 1990.
Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996.
Hans L. Bodlaender and Ton Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358-402, 1996.
Bostjan Bresar, Manoj Changat, Sandi Klavzar, Matjaz Kovse, Joseph Mathews, and Antony Mathews. Cover-incomparability graphs of posets. Order, 25(4):335-347, 2008.
Graham Brightwell and Peter Winkler. Counting linear extensions is #P-complete. In Proceedings of the Twenty-third Annual ACM Symposium on Theory of Computing, STOC'91, pages 175-181, 1991.
Russ Bubley and Martin Dyer. Faster random generation of linear extensions. Discrete Mathematics, 201(1-–3):81-88, 1999.
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Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
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Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
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http://dx.doi.org/10.1145/102782.102783
Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Inf. Comput., 209(2):143-153, 2011.
Stefan Felsner and Thibault Manneville. Linear extensions of N-free orders. Order, 32(2):147-155, 2014. URL: http://dx.doi.org/10.1007/s11083-014-9321-0.
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Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM J. Comput., 33(4):892-922, 2004.
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Thore Husfeldt, Ramamohan Paturi, Gregory B. Sorkin, and Ryan Williams. Exponential Algorithms: Algorithms and Complexity Beyond Polynomial Time (Dagstuhl Seminar 13331). Dagstuhl Reports, 3(8):40-72, 2013. URL: http://dx.doi.org/10.4230/DagRep.3.8.40.
http://dx.doi.org/10.4230/DagRep.3.8.40
Kustaa Kangas, Teemu Hankala, Teppo Niinimäki, and Mikko Koivisto. Counting linear extensions of sparse posets. In Proceedings of the 25th International Joint Conference on Artificial Intelligence, IJCAI 2016, New York City, USA, 2016. to appear.
T. Kloks. Treewidth: Computations and Approximations. Springer Verlag, Berlin, 1994.
Thomas Lukasiewicz, Maria Vanina Martinez, and Gerardo I. Simari. Probabilistic preference logic networks. In Torsten Schaub, Gerhard Friedrich, and Barry O'Sullivan, editors, ECAI 2014 - 21st European Conference on Artificial Intelligence, 18-22 August 2014, Prague, Czech Republic - Including Prestigious Applications of Intelligent Systems (PAIS 2014), volume 263 of Frontiers in Artificial Intelligence and Applications, pages 561-566. IOS Press, 2014.
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Rolf H. Möhring. Algorithms and Order, chapter Computationally Tractable Classes of Ordered Sets, pages 105-193. Springer Netherlands, 1989.
Jason Morton, Lior Pachter, Anne Shiu, Bernd Sturmfels, and Oliver Wienand. Convex rank tests and semigraphoids. SIAM Journal on Discrete Mathematics, 23(3):1117-1134, 2009. URL: http://dx.doi.org/10.1137/080715822.
http://dx.doi.org/10.1137/080715822
Teppo Mikael Niinimäki and Mikko Koivisto. Annealed importance sampling for structure learning in Bayesian networks. In IJCAI. IJCAI/AAAI, 2013.
Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for CNF formulas of bounded modular treewidth. Algorithmica, pages 1-27, 2015. URL: http://dx.doi.org/10.1007/s00453-015-0030-x.
http://dx.doi.org/10.1007/s00453-015-0030-x
Marcin Peczarski. New results in minimum-comparison sorting. Algorithmica, 40(2):133-145, July 2004. URL: http://dx.doi.org/10.1007/s00453-004-1100-7.
http://dx.doi.org/10.1007/s00453-004-1100-7
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A Constant Approximation Algorithm for Scheduling Packets on Line Networks
In this paper we improve the approximation ratio for the problem of scheduling packets on line networks with bounded buffers with the aim of maximizing the throughput. Each node in the network has a local buffer of bounded size B, and each edge (or link) can transmit a limited number c of packets in every time unit. The input to the problem consists of a set of packet requests, each defined by a source node, a destination node, and a release time. We denote by n the size of the network. A solution for this problem is a schedule that delivers (some of the) packets to their destinations without violating the capacity constraints of the network (buffers or edges). Our goal is to design an efficient algorithm that computes a schedule that maximizes the number of packets that arrive to their respective destinations.
We give a randomized approximation algorithm with constant approximation ratio for the case where the buffer-size to link-capacity ratio, B/c, does not depend on the input size. This improves over the previously best result of O(log^* n) [Räcke and Rosén SPAA 2009]. Our improvement is based on a new combinatorial lemma that we prove, stating, roughly speaking, that if packets are allowed to stay put in buffers only a limited number of time steps, 2d, where d is the longest source-destination distance, then the optimal solution is decreased by only a constant factor. This claim was not previously known in the integral (unsplitable, zero-one) case, and may find additional applications for routing and scheduling algorithms.
While we are not able to give the same improvement for the related problem when packets have hard deadlines, our algorithm does support "soft deadlines". That is, if packets have deadlines, we achieve a constant approximation ratio when the produced solution is allowed to miss deadlines by at most log n time units.
approximation algorithms
linear programming
randomized rounding
packet scheduling
admission control
40:1-40:16
Regular Paper
Guy
Even
Guy Even
Moti
Medina
Moti Medina
Adi
Rosén
Adi Rosén
10.4230/LIPIcs.ESA.2016.40
Micah Adler, Arnold L. Rosenberg, Ramesh K. Sitaraman, and Walter Unger. Scheduling time-constrained communication in linear networks. Theory Comput. Syst., 35(6):599-623, 2002. URL: http://dx.doi.org/10.1007/s00224-002-1001-6.
http://dx.doi.org/10.1007/s00224-002-1001-6
William Aiello, Eyal Kushilevitz, Rafail Ostrovsky, and Adi Rosén. Dynamic routing on networks with fixed-size buffers. In SODA, pages 771-780, 2003. URL: http://dx.doi.org/10.1145/644108.644236.
http://dx.doi.org/10.1145/644108.644236
Stanislav Angelov, Sanjeev Khanna, and Keshav Kunal. The network as a storage device: Dynamic routing with bounded buffers. Algorithmica, 55(1):71-94, 2009. (Appeared in APPROX-05). URL: http://dx.doi.org/10.1007/s00453-007-9143-1.
http://dx.doi.org/10.1007/s00453-007-9143-1
Baruch Awerbuch, Yossi Azar, and Amos Fiat. Packet routing via min-cost circuit routing. In ISTCS, pages 37-42, 1996.
Yossi Azar and Rafi Zachut. Packet routing and information gathering in lines, rings and trees. In ESA, pages 484-495, 2005. (See also manuscript in http://www.cs.tau.ac.il/~azar/). URL: http://dx.doi.org/10.1007/11561071_44.
http://dx.doi.org/10.1007/11561071_44
Guy Even and Moti Medina. An O(logn)-Competitive Online Centralized Randomized Packet-Routing Algorithm for Lines. In ICALP (2), pages 139-150, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14162-1_12.
http://dx.doi.org/10.1007/978-3-642-14162-1_12
Guy Even and Moti Medina. Online packet-routing in grids with bounded buffers. In Proc. 23rd Ann. ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pages 215-224, 2011. URL: http://dx.doi.org/10.1145/1989493.1989525.
http://dx.doi.org/10.1145/1989493.1989525
Guy Even and Moti Medina. Online packet-routing in grids with bounded buffers. CoRR, abs/1407.4498, 2014.
Guy Even, Moti Medina, and Boaz Patt-Shamir. Better deterministic online packet routing on grids. In Proceedings of the 27th ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2015, Portland, OR, USA, June 13-15, 2015, pages 284-293, 2015. URL: http://dx.doi.org/10.1145/2755573.2755588.
http://dx.doi.org/10.1145/2755573.2755588
Jon M Kleinberg. Approximation algorithms for disjoint paths problems. PhD thesis, Massachusetts Institute of Technology, 1996.
Harald Räcke and Adi Rosén. Approximation algorithms for time-constrained scheduling on line networks. In SPAA, pages 337-346, 2009. URL: http://dx.doi.org/10.1145/1583991.1584071.
http://dx.doi.org/10.1145/1583991.1584071
Prabhakar Raghavan. Randomized rounding and discrete ham-sandwich theorems: provably good algorithms for routing and packing problems. In Report UCB/CSD 87/312. Computer Science Division, University of California Berkeley, 1986.
Prabhakar Raghavan and Clark D Tompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365-374, 1987.
Adi Rosén and Gabriel Scalosub. Rate vs. buffer size-greedy information gathering on the line. ACM Transactions on Algorithms, 7(3):32, 2011. URL: http://dx.doi.org/10.1145/1978782.1978787.
http://dx.doi.org/10.1145/1978782.1978787
Neal E Young. Randomized rounding without solving the linear program. In SODA, volume 95, pages 170-178, 1995.
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Distributed Signaling Games
The study of the algorithmic and computational complexity of designing efficient signaling schemes for mechanisms aiming to optimize social welfare or revenue is a recurring theme in recent computer science literature. In reality, however, information is typically not held by a central authority, but is distributed among multiple sources (third-party "mediators"), a fact that dramatically changes the strategic and combinatorial nature of the signaling problem.
In this paper we introduce distributed signaling games, while using display advertising as a canonical example for introducing this foundational framework. A distributed signaling game may be a pure coordination game (i.e., a distributed optimization task), or a non-cooperative game. In the context of pure coordination games, we show a wide gap between the computational complexity of the centralized and distributed signaling problems, proving that distributed coordination on revenue-optimal signaling is a much harder problem than its "centralized" counterpart.
In the context of non-cooperative games, the outcome generated by the mediators' signals may have different value to each.
The reason for that is typically the desire of the auctioneer to align the incentives of the mediators with his own by a compensation relative to the marginal benefit from their signals. We design a mechanism for this problem via a novel application of Shapley's value, and show that it possesses a few interesting economical properties.
Signaling
display advertising
mechanism design
shapley value
41:1-41:16
Regular Paper
Moran
Feldman
Moran Feldman
Moshe
Tennenholtz
Moshe Tennenholtz
Omri
Weinstein
Omri Weinstein
10.4230/LIPIcs.ESA.2016.41
Robert J. Aumann. Agreeing to Disagree. The Annals of Statistics, 4(6):1236-1239, 1976.
Peter Bro Miltersen and Or Sheffet. Send mixed signals: Earn more, work less. In EC, pages 234-247, New York, NY, USA, 2012. ACM. URL: http://dx.doi.org/10.1145/2229012.2229033.
http://dx.doi.org/10.1145/2229012.2229033
Ruggiero Cavallo, R. Preston McAfee, and Sergei Vassilvitskii. Display advertising auctions with arbitrage. ACM Trans. Economics and Comput., 3(3):15, 2015. URL: http://dx.doi.org/10.1145/2668033.
http://dx.doi.org/10.1145/2668033
Yu Cheng, Ho Yee Cheung, Shaddin Dughmi, and Shang-Hua Teng. Signaling in quasipolynomial time. CoRR, abs/1410.3033, 2014. URL: http://arxiv.org/abs/1410.3033.
http://arxiv.org/abs/1410.3033
Shaddin Dughmi. On the hardness of signaling. In 55th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 354-363, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.45.
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Shaddin Dughmi, Nicole Immorlica, and Aaron Roth. Constrained signaling in auction design. In Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1341-1357, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.99.
http://dx.doi.org/10.1137/1.9781611973402.99
Yuval Emek, Michal Feldman, Iftah Gamzu, Renato Paes Leme, and Moshe Tennenholtz. Signaling schemes for revenue maximization. ACM Trans. Economics and Comput., 2(2):5, 2014. URL: http://dx.doi.org/10.1145/2594564.
http://dx.doi.org/10.1145/2594564
Eyal Even-Dar, Michael J. Kearns, and Jennifer Wortman. Sponsored search with contexts. In Internet and Network Economics, Third International Workshop (WINE), pages 312-317, 2007. URL: http://dx.doi.org/10.1007/978-3-540-77105-0_32.
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Moran Feldman, Moshe Tennenholtz, and Omri Weinstein. Distributed signaling games. CoRR, abs/1404.2861v2, 2015. URL: http://arxiv.org/abs/1404.2861v2.
http://arxiv.org/abs/1404.2861v2
Arpita Ghosh, Hamid Nazerzadeh, and Mukund Sundararajan. Computing optimal bundles for sponsored search. In Internet and Network Economics, Third International Workshop (WINE), pages 576-583, 2007. URL: http://dx.doi.org/10.1007/978-3-540-77105-0_63.
http://dx.doi.org/10.1007/978-3-540-77105-0_63
Mingyu Guo and Argyrios Deligkas. Revenue maximization via hiding item attributes. In IJCAI, 2013. URL: http://www.aaai.org/ocs/index.php/IJCAI/IJCAI13/paper/view/6909.
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Jonathan R. Mayer and John C. Mitchell. Third-party web tracking: Policy and technology. In IEEE Symposium on Security and Privacy (SP), pages 413-427, 2012. URL: http://dx.doi.org/10.1109/SP.2012.47.
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http://dx.doi.org/10.1145/1132516.1132528
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Shuai Yuan, Jun Wang, and Xiaoxue Zhao. Real-time bidding for online advertising: Measurement and analysis. In The Seventh International Workshop on Data Mining for Online Advertising (ADKDD), pages 3:1-3:8, 2013.
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New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness
We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is Omega(log^{1/2 - varepsilon} n), assuming NP not subseteq ZPTIME(n^{poly log n}). This constitutes a significant gap to the best known approximation upper bound of O(n^1/2) due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n/log log n) paths.
In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following.
- For MaxEDP, we give an O(r^0.5 log^1.5 kr)-approximation algorithm. As r<=n, up to logarithmic factors, our result strengthens the best known ratio O(n^0.5) due to Chekuri et al.
- Further, we show how to route Omega(opt) pairs with congestion O(log(kr)/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.
- For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)^O(r)n. This is a substantial improvement on the run time of 2^kr^O(r)n, which can be obtained via an algorithm by Scheffler.
We complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard for parameter r. This shows that neither problem is fixed-parameter tractable in r unless FPT = W[1] and that our approximability results are relevant even for very small constant values of r.
disjoint paths
approximation algorithms
feedback vertex set
42:1-42:17
Regular Paper
Krzysztof
Fleszar
Krzysztof Fleszar
Matthias
Mnich
Matthias Mnich
Joachim
Spoerhase
Joachim Spoerhase
10.4230/LIPIcs.ESA.2016.42
Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios Thilikos. Tight bounds for linkages in planar graphs. In Proc. ICALP 2011, volume 6755 of Lecture Notes Comput. Sci., pages 110-121, 2011.
Matthew Andrews. Approximation algorithms for the edge-disjoint paths problem via Räcke decompositions. In Proc. FOCS 2010, pages 277-286, 2010.
Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, and Lisa Zhang. Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica, 30(5):485-520, 2010.
Yonatan Aumann and Yuval Rabani. Improved bounds for all optical routing. In Proc. SODA 1995, pages 567-576, 1995.
Yonatan Aumann and Yuval Rabani. An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput., 27(1):291-301, 1998.
Baruch Awerbuch, Rainer Gawlick, Tom Leighton, and Yuval Rabani. On-line admission control and circuit routing for high performance computing and communication. In Proc. FOCS 1994, pages 412-423, 1994.
Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math., 12(3):289-297 (electronic), 1999.
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
http://dx.doi.org/10.1016/j.tcs.2011.04.039
Andrei Z. Broder, Alan M. Frieze, Stephen Suen, and Eli Upfal. Optimal construction of edge-disjoint paths in random graphs. SIAM J. Comput., 28(2):541-573 (electronic), 1999.
Andrei Z. Broder, Alan M. Frieze, and Eli Upfal. Existence and construction of edge-disjoint paths on expander graphs. SIAM J. Comput., 23(5):976-989, 1994.
Chandra Chekuri and Alina Ene. Poly-logarithmic approximation for maximum node disjoint paths with constant congestion. In Proc. SODA 2013, pages 326-341, 2013.
Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. An O(√n) approximation and integrality gap for disjoint paths and unsplittable flow. Theory Comput., 2:137-146, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a007.
http://dx.doi.org/10.4086/toc.2006.v002a007
Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. A note on multiflows and treewidth. Algorithmica, 54(3):400-412, 2009.
Chandra Chekuri, Marcelo Mydlarz, and F. Bruce Shepherd. Multicommodity demand flow in a tree and packing integer programs. ACM Trans. Algorithms, 3(3):Art. 27, 23, 2007.
Chandra Chekuri, Guyslain Naves, and F. Bruce Shepherd. Maximum edge-disjoint paths in k-sums of graphs. In Proc. ICALP 2013, volume 7965 of Lecture Notes Comput. Sci., pages 328-339, 2013.
Chandra Chekuri, Guyslain Naves, and F. Bruce Shepherd. Maximum edge-disjoint paths in k-sums of graphs, 2013. URL: http://arxiv.org/abs/1303.4897.
http://arxiv.org/abs/1303.4897
Chandra Chekuri, F. Bruce Shepherd, and Christophe Weibel. Flow-cut gaps for integer and fractional multiflows. J. Comb. Theory, Ser. B, 103(2):248-273, 2013.
Julia Chuzhoy. Routing in undirected graphs with constant congestion. In Proc. STOC 2012, pages 855-874, 2012.
Julia Chuzhoy, David H. K. Kim, and Shi Li. Improved approximation for node-disjoint paths in planar graphs. In Proc. STOC 2016, 2016. to appear.
Julia Chuzhoy and Shi Li. A polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2. In Proc. FOCS 2012, pages 233-242, 2012.
Alina Ene, Matthias Mnich, Marcin Pilipczuk, and Andrej Risteski. On routing disjoint paths in bounded treewidth graphs. In Proc. SWAT 2016, LIPIcs, 2016. to appear.
Krzysztof Fleszar, Matthias Mnich, and Joachim Spoerhase. New algorithms for maximum disjoint paths based on tree-likeness, 2016. URL: http://arxiv.org/abs/1603.01740.
http://arxiv.org/abs/1603.01740
Alan M. Frieze. Edge-disjoint paths in expander graphs. SIAM J. Comput., 30(6):1790-1801 (electronic), 2001.
Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica, 18(1):3-20, 1997. URL: http://dx.doi.org/10.1007/BF02523685.
http://dx.doi.org/10.1007/BF02523685
Oktay Günlük. A new min-cut max-flow ratio for multicommodity flows. SIAM J. Discrete Math., 21(1):1-15, 2007.
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Richard M. Karp. On the computational complexity of combinatorial problems. Networks, 5:45-68, 1975.
Ken-ichi Kawarabayashi and Yusuke Kobayashi. Breaking O(n^1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two. In Proc. STOC 2011, pages 81-88, 2011.
Ken-ichi Kawarabayashi and Paul Wollan. A shorter proof of the graph minor algorithm: the unique linkage theorem. In Proc. STOC 2010, pages 687-694, 2010.
Jon Kleinberg and Ronitt Rubinfeld. Short paths in expander graphs. In Proc. FOCS 1996, pages 86-95, 1996.
Jon Kleinberg and Éva Tardos. Disjoint paths in densely embedded graphs. In Proc. FOCS 1995, pages 52-61, 1995.
Jon Kleinberg and Éva Tardos. Approximations for the disjoint paths problem in high-diameter planar networks. J. Comput. System Sci., 57(1):61-73, 1998.
Stavros G. Kolliopoulos and Clifford Stein. Approximating disjoint-path problems using packing integer programs. Math. Program., 99(1):63-87, 2004.
Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999.
Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995.
Daniel Lokshtanov, M. S. Ramanujan, and Saket Saurabh. Linear time parameterized algorithms for subset feedback vertex set. In Proc. ICALP 2015, pages 935-946, 2015.
Takao Nishizeki, Jens Vygen, and Xiao Zhou. The edge-disjoint paths problem is NP-complete for series-parallel graphs. Discrete Appl. Math., 115(1-3):177-186, 2001. URL: http://dx.doi.org/10.1016/S0166-218X(01)00223-2.
http://dx.doi.org/10.1016/S0166-218X(01)00223-2
Prabhakar Raghavan and Clark D. Tompson. Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365-374, 1987.
Satish Rao and Shuheng Zhou. Edge disjoint paths in moderately connected graphs. SIAM J. Comput., 39(5):1856-1887, 2010.
Neil Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65-110, 1995. URL: http://dx.doi.org/10.1006/jctb.1995.1006.
http://dx.doi.org/10.1006/jctb.1995.1006
Petra Scheffler. A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Technical Report TR 396/1994, FU Berlin, Fachbereich 3 Mathematik, 1994.
Loïc Séguin-Charbonneau and F. Bruce Shepherd. Maximum edge-disjoint paths in planar graphs with congestion 2. In Proc. FOCS 2011, pages 200-209, 2011.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Streaming Property Testing of Visibly Pushdown Languages
In the context of formal language recognition, we demonstrate the superiority of streaming property testers against streaming algorithms and property testers, when they are not combined. Initiated by Feigenbaum et al., a streaming property tester is a streaming algorithm recognizing a language under the property testing approximation: it must distinguish inputs of the language from those that are eps-far from it, while using the smallest possible memory (rather than limiting its number of input queries). Our main result is a streaming eps-property tester for visibly pushdown languages (V_{PL}) with memory space poly(log n /epsilon).
Our construction is done in three steps. First, we simulate a visibly pushdown automaton in one pass using a stack of small height but whose items can be of linear size. In a second step, those items are replaced by small sketches. Those sketches rely on a notion of suffix-sampling we introduce. This sampling is the key idea for taking benefit of both streaming algorithms and property testers in the third step. Indeed, the last step relies on a (non-streaming) property tester for weighted regular languages based on a previous tester by Alon et al. This tester can directly be used for streaming testing special cases of instances of V_{PL} that are already hard for both streaming algorithms and property testers. We then use it to decide the correctness of completed items, given their sketches, before removing them from the stack.
Streaming Algorithm
Property Testing
Visibly Pushdown Languages
43:1-43:17
Regular Paper
Nathanaël
François
Nathanaël François
Frédéric
Magniez
Frédéric Magniez
Michel
de Rougemont
Michel de Rougemont
Olivier
Serre
Olivier Serre
10.4230/LIPIcs.ESA.2016.43
N. Alon, M. Krivelevich, I. Newman, and M. Szegedy. Regular languages are testable with a constant number of queries. SIAM Journal on Computing, 30(6), 2000.
N. Alon, Y. Matias, and M. Szegedy. The space complexity of approximating the frequency moments. Journal of Computer and System Sciences, 58(1):137-147, 1999.
R. Alur. Marrying words and trees. In Proc. of 26th ACM Symposium on Principles of Database Systems, pages 233-242, 2007.
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Streaming Pattern Matching with d Wildcards
In the pattern matching with d wildcards problem we are given a text T of length n and a pattern P of length m that contains d wildcard characters, each denoted by a special symbol '?'. A wildcard character matches any other character. The goal is to establish for each m-length substring of T whether it matches P. In the streaming model variant of the pattern matching with d wildcards problem the text T arrives one character at a time and the goal is to report, before the next character arrives, if the last m characters match P while using only o(m) words of space.
In this paper we introduce two new algorithms for the d wildcard pattern matching problem in the streaming model.
The first is a randomized Monte Carlo algorithm that is parameterized by a constant 0<=delta<=1. This algorithm uses ~O(d^{1-delta}) amortized time per character and ~O(d^{1+delta}) words of space. The second algorithm, which is used as a black box in the first algorithm, is a randomized Monte Carlo algorithm which uses O(d+log m) worst-case time per character and O(d log m) words of space.
wildcards
don't-cares
streaming pattern matching
fingerprints
44:1-44:16
Regular Paper
Shay
Golan
Shay Golan
Tsvi
Kopelowitz
Tsvi Kopelowitz
Ely
Porat
Ely Porat
10.4230/LIPIcs.ESA.2016.44
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. Syst. Sci., 58(1):137-147, 1999. URL: http://dx.doi.org/10.1006/jcss.1997.1545.
http://dx.doi.org/10.1006/jcss.1997.1545
Amihood Amir, Moshe Lewenstein, and Ely Porat. Faster algorithms for string matching with k mismatches. J. Algorithms, 50(2):257-275, 2004. URL: http://dx.doi.org/10.1016/S0196-6774(03)00097-X.
http://dx.doi.org/10.1016/S0196-6774(03)00097-X
Jean Berstel and Luc Boasson. Partial words and a theorem of fine and wilf. Theor. Comput. Sci., 218(1):135-141, 1999. URL: http://dx.doi.org/10.1016/S0304-3975(98)00255-2.
http://dx.doi.org/10.1016/S0304-3975(98)00255-2
Francine Blanchet-Sadri. Algorithmic Combinatorics on Partial Words. Discrete mathematics and its applications. CRC Press, 2008. URL: http://www.crcpress.com/product/isbn/9781420060928.
http://www.crcpress.com/product/isbn/9781420060928
Francine Blanchet-Sadri and Robert A. Hegstrom. Partial words and a theorem of fine and wilf revisited. Theor. Comput. Sci., 270(1-2):401-419, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(00)00407-2.
http://dx.doi.org/10.1016/S0304-3975(00)00407-2
Dany Breslauer and Zvi Galil. Real-time streaming string-matching. ACM Transactions on Algorithms, 10(4):22:1-22:12, 2014. URL: http://dx.doi.org/10.1145/2635814.
http://dx.doi.org/10.1145/2635814
Dany Breslauer, Roberto Grossi, and Filippo Mignosi. Simple real-time constant-space string matching. Theor. Comput. Sci., 483:2-9, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2012.11.040.
http://dx.doi.org/10.1016/j.tcs.2012.11.040
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Peter Clifford and Raphaël Clifford. Simple deterministic wildcard matching. Inf. Process. Lett., 101(2):53-54, 2007. URL: http://dx.doi.org/10.1016/j.ipl.2006.08.002.
http://dx.doi.org/10.1016/j.ipl.2006.08.002
Raphaël Clifford, Klim Efremenko, Benny Porat, and Ely Porat. A black box for online approximate pattern matching. Inf. Comput., 209(4):731-736, 2011. URL: http://dx.doi.org/10.1016/j.ic.2010.12.007.
http://dx.doi.org/10.1016/j.ic.2010.12.007
Raphaël Clifford, Klim Efremenko, Ely Porat, and Amir Rothschild. From coding theory to efficient pattern matching. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 778-784, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496855.
http://dl.acm.org/citation.cfm?id=1496770.1496855
Raphaël Clifford, Klim Efremenko, Ely Porat, and Amir Rothschild. Pattern matching with don't cares and few errors. J. Comput. Syst. Sci., 76(2):115-124, 2010. URL: http://dx.doi.org/10.1016/j.jcss.2009.06.002.
http://dx.doi.org/10.1016/j.jcss.2009.06.002
Raphaël Clifford, Allyx Fontaine, Ely Porat, Benjamin Sach, and Tatiana A. Starikovskaya. Dictionary matching in a stream. In Nikhil Bansal and Irene Finocchi, editors, Proc. 23rd Annual European Symposium on Algorithms (ESA'15), volume 9294 of LNCS, pages 361-372. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_31.
http://dx.doi.org/10.1007/978-3-662-48350-3_31
Raphaël Clifford, Allyx Fontaine, Ely Porat, Benjamin Sach, and Tatiana A. Starikovskaya. The k-mismatch problem revisited. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2039-2052, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch142.
http://dx.doi.org/10.1137/1.9781611974331.ch142
Raphaël Clifford, Markus Jalsenius, Ely Porat, and Benjamin Sach. Space lower bounds for online pattern matching. Theor. Comput. Sci., 483:68-74, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2012.06.012.
http://dx.doi.org/10.1016/j.tcs.2012.06.012
Raphaël Clifford and Ely Porat. A filtering algorithm for k-mismatch with don't cares. In String Processing and Information Retrieval, 14th International Symposium, SPIRE 2007, Santiago, Chile, October 29-31, 2007, Proceedings, pages 130-136, 2007. URL: http://dx.doi.org/10.1007/978-3-540-75530-2_12.
http://dx.doi.org/10.1007/978-3-540-75530-2_12
Raphaël Clifford and Benjamin Sach. Pseudo-realtime pattern matching: Closing the gap. In Amihood Amir and Laxmi Parida, editors, Combinatorial Pattern Matching, 21st Annual Symposium, CPM 2010, New York, NY, USA, June 21-23, 2010. Proceedings, volume 6129 of Lecture Notes in Computer Science, pages 101-111. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13509-5_10.
http://dx.doi.org/10.1007/978-3-642-13509-5_10
Richard Cole and Ramesh Hariharan. Verifying candidate matches in sparse and wildcard matching. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 592-601. ACM, 2002. URL: http://dx.doi.org/10.1145/509907.509992.
http://dx.doi.org/10.1145/509907.509992
Funda Ergün, Hossein Jowhari, and Mert Saglam. Periodicity in streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, Barcelona, Spain, September 1-3, 2010. Proceedings, pages 545-559, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15369-3_41.
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Shay Golan, Tsvi Kopelowitz, and Ely Porat. Streaming pattern matching with d wildcards. CoRR, abs/1605.16729, 2015. URL: http://arxiv.org/abs/1605.16729.
http://arxiv.org/abs/1605.16729
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http://dx.doi.org/10.1561/0400000002
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http://dx.doi.org/10.1007/978-3-540-73437-6_19
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http://dx.doi.org/10.1016/j.tcs.2009.07.010
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How Hard is it to Find (Honest) Witnesses?
In recent years much effort has been put into developing polynomial-time conditional lower bounds for algorithms and data structures in both static and dynamic settings. Along these lines we introduce a framework for proving conditional lower bounds based on the well-known 3SUM conjecture. Our framework creates a compact representation of an instance of the 3SUM problem using hashing and domain specific encoding. This compact representation admits false solutions to the original 3SUM problem instance which we reveal and eliminate until we find a true solution. In other words, from all witnesses (candidate solutions) we figure out if an honest one (a true solution) exists. This enumeration of witnesses is used to prove conditional lower bounds on reporting problems that generate all witnesses. In turn, these reporting problems are then reduced to various decision problems using special search data structures which are able to enumerate the witnesses while only using solutions to decision variants. Hence, 3SUM-hardness of the decision problems is deduced.
We utilize this framework to show conditional lower bounds for several variants of convolutions, matrix multiplication and string problems. Our framework uses a strong connection between all of these problems and the ability to find witnesses.
Specifically, we prove conditional lower bounds for computing partial outputs of convolutions and matrix multiplication for sparse inputs. These problems are inspired by the open question raised by Muthukrishnan 20 years ago. The lower bounds we show rule out the possibility (unless the 3SUM conjecture is false) that almost linear time solutions to sparse input-output convolutions or matrix multiplications exist. This is in contrast to standard convolutions and matrix multiplications that have, or assumed to have, almost linear solutions.
Moreover, we improve upon the conditional lower bounds of Amir et al. for histogram indexing, a problem that has been of much interest recently. The conditional lower bounds we show apply for both reporting and decision variants. For the well-studied decision variant, we show a full tradeoff between preprocessing and query time for every alphabet size > 2. At an extreme, this implies that no solution to this problem exists with subquadratic preprocessing time and ~O(1) query time for every alphabet size > 2, unless the 3SUM conjecture is false. This is in contrast to a recent result by Chan and Lewenstein for a binary alphabet.
While these specific applications are used to demonstrate the techniques of our framework, we believe that this novel framework is useful for many other problems as well.
3SUM
convolutions
matrix multiplication
histogram indexing
45:1-45:16
Regular Paper
Isaac
Goldstein
Isaac Goldstein
Tsvi
Kopelowitz
Tsvi Kopelowitz
Moshe
Lewenstein
Moshe Lewenstein
Ely
Porat
Ely Porat
10.4230/LIPIcs.ESA.2016.45
Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-sum conjecture. In Int'l Colloquium on Automata, Languages and Programming, ICALP 2013, pages 1-12, 2013.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Foundations of Computer Science, FOCS 2014, pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In International Colloquium on Automata, Languages and Programming, ICALP 2014, pages 39-51, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Symposium on Theory of Computing, STOC 2015, pages 41-50, 2015.
Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In International Colloquium on Automata, Languages and Programming, ICALP 2014, pages 114-125, 2014.
Ilya Baran, Erik D. Demaine, and Mihai Patrascu. Subquadratic algorithms for 3SUM. In Workshop on Algorithms and Data Structures, WADS 2005, pages 409-421, 2005.
David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Patrascu, and Perouz Taslakian. Necklaces, convolutions, and X+Y. Algorithmica, 69(2):294-314, 2014.
Peter Burcsi, Ferdinando Cicalese, Gabriele Fici, and Zsuzsanna Lipták. Algorithms for jumbled pattern matching in strings. Int. J. Found. Comput. Sci., 23(2):357-374, 2012.
Timothy Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Symposium on Theory of Computing, STOC 2015, pages 31-40, 2015.
Ferdinando Cicalese, Gabriele Fici, and Zsuzsanna Lipták. Searching for jumbled patterns in strings. In Prague Stringology Conference, pages 105-117, 2009.
Richard Cole and Ramesh Hariharan. Verifying candidate matches in sparse and wildcard matching. In Symposium on Theory of Computing, STOC 2002, pages 592-601, 2002.
Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In Symposium on Theoretical Aspects of Computer Science, STACS 1996, pages 569-580, 1996.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Comput. Geom., 5:165-185, 1995.
François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation, ISSAC 2014, pages 296-303, 2014.
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. Nearly optimal sparse fourier transform. In Symposium on Theory of Computing Conference, STOC 2012, pages 563-578, 2012.
Danny Hermelin, Gad M. Landau, Yuri Rabinovich, and Oren Weimann. Binary jumbled pattern matching via all-pairs shortest paths. CoRR, abs/1401.2065, 2014.
Allan Grønlund Jørgensen and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science, FOCS 2014, pages 621-630, 2014.
Tomasz Kociumaka, Jakub Radoszewski, and Wojciech Rytter. Efficient indexes for jumbled pattern matching with constant-sized alphabet. In ESA, pages 625-636, 2013.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3SUM conjecture. In Symposium on Discrete Algorithms, SODA 2016, pages 1272-1287, 2016.
Tanaeem M. Moosa and M. Sohel Rahman. Indexing permutations for binary strings. Inf. Process. Lett., 110(18-19):795-798, 2010.
Tanaeem M. Moosa and M. Sohel Rahman. Sub-quadratic time and linear space data structures for permutation matching in binary strings. J. Discrete Algorithms, 10:5-9, 2012.
S. Muthukrishnan. New results and open problems related to non-standard stringology. In CPM 1995, pages 298-317, 1995.
Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Symposium on Theory of Computing Conference, STOC 2010, pages 603-610, 2010.
Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Symposium on Theory of Computing, STOC 2014, pages 664-673, 2014.
Virginia Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Symposium on Theory of Computing Conference, STOC 2012, pages 887-898, 2012.
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Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time
We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with ~O(1) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997].
We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(n log n/epsilon^2) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+epsilon)-approximation to the minimum cut. The algorithm has ~O(1) amortized update-time and constant query-time.
Dynamic Graph Algorithms
Minimum Cut
Edge Connectivity
46:1-46:17
Regular Paper
Gramoz
Goranci
Gramoz Goranci
Monika
Henzinger
Monika Henzinger
Mikkel
Thorup
Mikkel Thorup
10.4230/LIPIcs.ESA.2016.46
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. of the 55th FOCS, pages 434-443. IEEE, 2014.
Kook Jin Ahn and Sudipto Guha. Graph sparsification in the semi-streaming model. In Proc. of the 36th ICALP, pages 328-338, 2009.
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proc. of the 32nd PODS, pages 5-14, 2012.
András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015.
Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Charalampos E. Tsourakakis. Space- and time-efficient algorithm for maintaining dense subgraphs on one-pass dynamic streams. In Proc. of the 47th STOC, pages 173-182, 2015.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms (3. ed.). MIT Press, 2009.
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Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proc. of the 47th STOC, pages 21-30, 2015.
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Danupon Nanongkai and Thatchaphol Saranurak. Dynamic cut oracle. under submission, 2016.
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Packing and Covering with Non-Piercing Regions
In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). We show that the local
search algorithm that yields PTASs when the regions are disks [Aschner/Katz/Morgenstern/Yuditsky, WALCOM 2013; Gibson/Pirwani, 2005; Mustafa/Raman/Ray, 2015] can be extended to work for non-piercing regions. While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks. The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks. In order to prove our results, we introduce novel techniques that we believe will find applications in other problems.
We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded. Our result settles a conjecture of Har-Peled (see Conclusion of [Har-Peled, SoCG 2014]) in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane.
Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our
objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al. [Ene/Har-Peled/Raichel, SoCG 2012].
Local Search
Set Cover
Dominating Set
Capacitated Packing
Approximation algorithms
47:1-47:17
Regular Paper
Sathish
Govindarajan
Sathish Govindarajan
Rajiv
Raman
Rajiv Raman
Saurabh
Ray
Saurabh Ray
Aniket
Basu Roy
Aniket Basu Roy
10.4230/LIPIcs.ESA.2016.47
Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS'13, pages 400-409, Washington, DC, USA, 2013. IEEE Computer Society.
Anna Adamaszek and Andreas Wiese. A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'14, pages 645-656, 2014.
Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Computational Geometry, 34(2):83-95, 2006.
Pankaj K. Agarwal, Marc van Kreveld, and Subhash Suri. Label placement by maximum independent set in rectangles. Computational Geometry, 11(3):209-218, 1998.
Rom Aschner, Matthew J. Katz, Gila Morgenstern, and Yelena Yuditsky. Approximation schemes for covering and packing. In WALCOM: Algorithms and Computation, volume 7748 of Lecture Notes in Computer Science, pages 89-100. Springer Berlin Heidelberg, 2013.
Franz Aurenhammer. Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Computing Surveys (CSUR), 23(3):345-405, 1991.
Reuven Bar-Yehuda, Danny Hermelin, and Dror Rawitz. Minimum vertex cover in rectangle graphs. Computational Geometry, 44(6):356-364, 2011.
Vijay V. S. P. Bhattiprolu and Sariel Har-Peled. Separating a Voronoi Diagram via Local Search. In Proceedings of the Thirty-second International Symposium on Computational Geometry, SoCG'16, pages 18:1-18:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
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Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms, 46(2):178-189, 2003.
Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'12, pages 1576-1585, 2012.
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012.
Vincent Cohen-Addad and Claire Mathieu. Effectiveness of local search for geometric optimization. In Proceedings of the Thirty-first International Symposium on Computational Geometry, SoCG'15, pages 329-343, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Jeffrey S. Doerschler and Herbert Freeman. A rule-based system for dense-map name placement. Communications of the ACM, 35(1):68-79, 1992.
Stephane Durocher and Robert Fraser. Duality for geometric set cover and geometric hitting set problems on pseudodisks. In Proceedings of the 27th Canadian Conference on Computational Geometry, pages 8-16, 2015.
Alina Ene, Sariel Har-Peled, and Benjamin Raichel. Geometric packing under non-uniform constraints. In Proceedings of the Twenty-eighth Annual Symposium on Computational Geometry, SoCG'12, pages 11-20, New York, NY, USA, 2012. ACM.
Thomas Erlebach and Erik Jan van Leeuwen. Approximating geometric coverage problems. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete algorithms, SODA'08, pages 1267-1276, 2008.
Guy Even, Dror Rawitz, and Shimon (Moni) Shahar. Hitting sets when the VC-dimension is small. Inf. Process. Lett., 95(2):358-362, July 2005.
Matt Gibson and Imran A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the log n barrier. In Algorithms - ESA 2010 - 18th Annual European Symposium, Liverpool, United Kingdom, September 6-8, 2010, Proceedings, pages 243-254, 2010.
Sariel Har-Peled. Quasi-polynomial time approximation scheme for sparse subsets of polygons. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SoCG'14, pages 120-129, New York, NY, USA, 2014. ACM.
Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 717-728, 2015.
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Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi Varadarajan. Guarding terrains via local search. Journal of Computational Geometry, 5(1):168-178, 2014.
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Nabil H. Mustafa, Rajiv Raman, and Saurabh Ray. Quasi-polynomial time approximation scheme for weighted geometric set cover on pseudodisks and halfspaces. SIAM Journal on Computing, 44(6):1650-1669, 2015.
Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010.
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Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size d; after the update we have to answer queries.
In this paper, we consider the dynamic subgraph connectivity problem with sensitivity d: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to $d$ vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices.
We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice.
connectivity
emergency planning
sensitivity
48:1-48:11
Regular Paper
Monika
Henzinger
Monika Henzinger
Stefan
Neumann
Stefan Neumann
10.4230/LIPIcs.ESA.2016.48
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pages 434-443, Philadelphia, PA, USA, 2014.
Surender Baswana, Shreejit Ray Chaudhury, Keerti Choudhary, and Shahbaz Khan. Dynamic DFS tree in undirected graphs: breaking the O(m) barrier. CoRR, abs/1502.02481, 2015.
Aaron Bernstein and David Karger. Improved distance sensitivity oracles via random sampling. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 34-43, Philadelphia, PA, USA, 2008.
Aaron Bernstein and David Karger. A nearly optimal oracle for avoiding failed vertices and edges. In Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing (STOC), pages 101-110, New York, NY, USA, 2009.
Timothy M. Chan. Dynamic subgraph connectivity with geometric applications. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing (STOC), pages 7-13, New York, NY, USA, 2002.
Timothy M Chan, Mihai Patrascu, and Liam Roditty. Dynamic connectivity: Connecting to networks and geometry. SIAM Journal on Computing, 40(2):333-349, 2011.
Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-sensitivity distance oracles and routing schemes. Algorithmica, 63(4):861-882, 2011.
Camil Demetrescu, Mikkel Thorup, Rezaul Alam Chowdhury, and Vijaya Ramachandran. Oracles for distances avoiding a failed node or link. SIAM Journal on Computing, 37(5):1299-1318, 2008.
Ran Duan. New data structures for subgraph connectivity. In 37th International Colloquium on Automata, Languages and Programming (ICALP), pages 201-212, Bordeaux, France, 2010.
Ran Duan and Seth Pettie. Dual-failure distance and connectivity oracles. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 506-515, Philadelphia, PA, USA, 2009.
Ran Duan and Seth Pettie. Connectivity oracles for failure prone graphs. In Proceedings of the Forty-second ACM Symposium on Theory of Computing (STOC), pages 465-474, New York, NY, USA, 2010.
David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. Journal of the ACM, 44(5):669-696, 1997.
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D. Frigioni and F. G. Italiano. Dynamically switching vertices in planar graphs. Algorithmica, 28(1):76-103, 2000.
David Gibb, Bruce M. Kapron, Valerie King, and Nolan Thorn. Dynamic graph connectivity with improved worst case update time and sublinear space. CoRR, abs/1509.06464, 2015.
Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), pages 21-30, New York, NY, USA, 2015.
Monika R. Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM, 46(4):502-516, 1999.
Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM, 48(4):723-760, 2001.
Bruce M. Kapron, Valerie King, and Ben Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1131-1142, 2013.
Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup. Faster worst case deterministic dynamic connectivity. CoRR, abs/1507.05944, 2015.
Neelesh Khanna and Surender Baswana. Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science (STACS), Leibniz International Proceedings in Informatics (LIPIcs), pages 513-524, 2010.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1272-1287, 2016.
Mihai Patrascu and Mikkel Thorup. Planning for fast connectivity updates. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 263-271, 2007.
Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing (STOC), pages 343-350, 2000.
Christian Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1757-1769, 2013.
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A Combinatorial Approximation Algorithm for Graph Balancing with Light Hyper Edges
Makespan minimization in restricted assignment (R|p_{ij} in {p_j, infinity}|C_{max}) is a classical problem in the field of machine scheduling. In a landmark paper, [Lenstra, Shmoys, and Tardos, Math. Progr. 1990] gave a 2-approximation algorithm and proved that the problem cannot be approximated within 1.5 unless P=NP. The upper and lower bounds of the problem have been essentially unimproved in the intervening 25 years, despite several remarkable successful attempts in some special cases of the problem recently.
In this paper, we consider a special case called graph-balancing with light hyper edges, where heavy jobs can be assigned to at most two machines while light jobs can be assigned to any number of machines. For this case, we present algorithms with approximation ratios strictly better than 2. Specifically,
- Two job sizes: Suppose that light jobs have weight w and heavy jobs have weight W, and w < W. We give a 1.5-approximation algorithm (note that the current 1.5 lower bound is established in an even more restrictive setting). Indeed, depending on the specific values of w and W, sometimes our algorithm guarantees sub-1.5 approximation ratios.
- Arbitrary job sizes: Suppose that W is the largest given weight, heavy jobs have weights in the range of (beta W, W], where 4/7 <= beta < 1, and light jobs have weights in the range of (0,beta W]. We present a (5/3+beta/3)-approximation algorithm.
Our algorithms are purely combinatorial, without the need of solving a linear program as required in most other known approaches.
Approximation Algorithms
Machine Scheduling
Graph Balancing
Combinatorial Algorithms
49:1-49:15
Regular Paper
Chien-Chung
Huang
Chien-Chung Huang
Sebastian
Ott
Sebastian Ott
10.4230/LIPIcs.ESA.2016.49
Yuichi Asahiro, Jesper Jansson, Eiji Miyano, Hirotaka Ono, and Kouhei Zenmyo. Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. J. Comb. Optim., 22(1):78-96, 2011.
Deeparnab Chakrabarty, Sanjeev Khanna, and Shi Li. On (1,ε)-restricted assignment makespan minimization. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15), pages 1087-1101. SIAM, 2015.
T. Ebenlendr, M. Krčál, and J. Sgall. Graph balancing: A special case of scheduling unrelated parallel machines. Algorithmica, 68:62-80, 2014.
Tomáš Ebenlendr, Marek Krčál, and Jiří Sgall. Graph balancing: A special case of scheduling unrelated parallel machines. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'08), pages 483-490. SIAM, 2008.
M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. Computing nash equilibria for scheduling on restricted parallel links. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC'04), pages 613-622. ACM, 2004.
M. Gairing, B. Monien, and A. Wocalw. A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theor. Comput. Sci., 380(1-2):87-99, 2007.
Chien-Chung Huang and Sebastian Ott. A combinatorial approximation algorithm for graph balancing with light hyper edges. CoRR, abs/1507.07396, 2015.
S. G. Kolliopoulos and Y. Moysoglou. The 2-valued case of makespan minimization with assignment constraints. Information Processing Letters, 113(1-2):39-43, 2013.
J.K. Lenstra, D.B. Shmoys, and É. Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46:256-271, 1990.
J.Y. Leung and C. Li. Scheduling with processing set restrictions: A survey. International Journal of Production Economics, 116:251-262, 2008.
J. Sgall. Private communication, 2015.
E. V. Shchepin and N. Vakhania. An optimal rounding gives a better approximation for scheduling unrelated machines. Oper. Res. Lett., 33(2):127-133, 2005.
O. Svensson. Santa claus schedules jobs on unrelated machines. SIAM J. Comput., 41(5):1318-1341, 2012.
J. Verschae and A. Wiese. On the configuration-lp for scheduling on unrelated machines. J. Scheduling, 17(4):371-383, 2014.
D. P. Willamson and D.B. Shmoys. The design of approximation algorithms. Cambridge University Press, 2010.
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epsilon-Kernel Coresets for Stochastic Points
With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an epsilon-EXP-KERNEL), as well as the probability distribution on the width (an (epsilon, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(epsilon^{-(d-1)/2}) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an epsilon-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (epsilon, tau)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty.
e-kernel
coreset
stochastic point
shape fitting
50:1-50:18
Regular Paper
Lingxiao
Huang
Lingxiao Huang
Jian
Li
Jian Li
Jeff M.
Phillips
Jeff M. Phillips
Haitao
Wang
Haitao Wang
10.4230/LIPIcs.ESA.2016.50
A. Abdullah, S. Daruki, and J.M. Phillips. Range counting coresets for uncertain data. In Proceedings 29th ACM Syposium on Computational Geometry, pages 223-232, 2013.
Marcel R Ackermann, Johannes Blömer, and Christian Sohler. Clustering for metric and nonmetric distance measures. ACM Transactions on Algorithms (TALG), 6(4):59, 2010.
P. Afshani, P.K. Agarwal, L. Arge, K.G. Larsen, and J.M. Phillips. (Approximate) uncertain skylines. In Proceedings of the 14th International Conference on Database Theory, pages 186-196, 2011.
P.K. Agarwal, S.-W. Cheng, and K. Yi. Range searching on uncertain data. ACM Transactions on Algorithms (TALG), 8(4):43, 2012.
P.K. Agarwal, A. Efrat, S. Sankararaman, and W. Zhang. Nearest-neighbor searching under uncertainty. In Proceedings of the 31st Symposium on Principles of Database Systems, pages 225-236, 2012.
P.K. Agarwal, S. Har-Peled, S. Suri, H. Yıldız, and W. Zhang. Convex hulls under uncertainty. In Proceedings of the 22nd Annual European Symposium on Algorithms, pages 37-48, 2014.
P.K. Agarwal, S. Har-Peled, and K.R. Varadarajan. Approximating extent measures of points. Journal of the ACM, 51(4):606-635, 2004.
P.K. Agarwal, S. Har-Peled, and K.R. Varadarajan. Geometric approximation via coresets. Combinatorial and Computational Geometry, 52:1-30, 2005.
P.K. Agarwal and M. Sharir. Arrangements and their applications. Handbook of Computational Geometry, J. Sack and J. Urrutia (eds.), pages 49-119. Elsevier, Amsterdam, The Netherlands, 2000.
Martin Anthony and Peter L Bartlett. Neural network learning: Theoretical foundations. cambridge university press, 2009.
D. Bandyopadhyay and J. Snoeyink. Almost-Delaunay simplices: Nearest neighbor relations for imprecise points. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, pages 410-419, 2004.
Saugata Basu, Richard Pollack, and M Roy. Algorithms in real algebraic geometry. AMC, 10:12, 2011.
T.M. Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry: Theory and Applications, 35:20-35, 2006.
K. Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009.
R. Cheng, J. Chen, and X. Xie. Cleaning uncertain data with quality guarantees. Proceedings of the VLDB Endowment, 1(1):722-735, 2008.
G. Cormode and A. McGregor. Approximation algorithms for clustering uncertain data. In Proceedings of the 27th Symposium on Principles of Database Systems, pages 191-200, 2008.
A. Deshpande, L. Rademacher, S. Vempala, and G. Wang. Matrix approximation and projective clustering via volume sampling. In Proceedings of the 17th ACM-SIAM symposium on Discrete algorithm, pages 1117-1126, 2006.
X. Dong, A.Y. Halevy, and C. Yu. Data integration with uncertainty. In Proceedings of the 33rd International Conference on Very Large Data Bases, pages 687-698, 2007.
A. Driemel, H. HAverkort, M. Löffler, and R.I. Silveira. Flow computations on imprecise terrains. Journal of Computational Geometry, 4:38-78, 2013.
W. Evans and J. Sember. The possible hull of imprecise points. In Proceedings of the 23rd Canadian Conference on Computational Geometry, 2011.
D. Feldman, A. Fiat, H. Kaplan, and K. Nissim. Private coresets. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 361-370, 2009.
D. Feldman and M. Langberg. A unified framework for approximating and clustering data. In Proceedings of the 43rd ACM Symposium on Theory of Computing, pages 569-578, 2011.
Dan Feldman and Leonard J Schulman. Data reduction for weighted and outlier-resistant clustering. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1343-1354. SIAM, 2012.
Martin Fink, John Hershberger, Nirman Kumar, and Subhash Suri. Hyperplane seperability and convexity of probabilistic point sets. In Proceedings Symposium on Computational Geometry, 2016.
P.K. Ghosh and K.V. Kumar. Support function representation of convex bodies, its application in geometric computing, and some related representations. Computer Vision and Image Understanding, 72(3):379-403, 1998.
S. Guha and K. Munagala. Exceeding expectations and clustering uncertain data. In Proceedings of the 28th Symposium on Principles of Database Systems, pages 269-278, 2009.
L.J. Guibas, D. Salesin, and J. Stolfi. Constructing strongly convex approximate hulls with inaccurate primitives. Algorithmica, 9:534-560, 1993.
S. Har-Peled. On the expected complexity of random convex hulls. arXiv:1111.5340, 2011.
S. Har-Peled and S. Mazumdar. On coresets for k-means and k-median clustering. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 291-300, 2004.
Sariel Har-Peled and Yusu Wang. Shape fitting with outliers. SIAM Journal on Computing, 33(2):269-285, 2004.
M. Held and J.S.B. Mitchell. Triangulating input-constrained planar point sets. Information Processing Letters, 109(1):54-56, 2008.
Lingxiao Huang and Jian Li. Approximating the expected values for combinatorial optimization problems over stochastic points. In The 42nd International Colloquium on Automata, Languages, and Programming, pages 910-921. Springer, 2015.
A.G. Jørgensen, M. Löffler, and J.M. Phillips. Geometric computation on indecisive points. In Proceedings of the 12th Algorithms and Data Structure Symposium, pages 536-547, 2011.
P. Kamousi, T.M. Chan, and S. Suri. The stochastic closest pair problem and nearest neighbor search. In Proceedings of the 12th Algorithms and Data Structure Symposium, pages 548-559, 2011.
P. Kamousi, T.M. Chan, and S. Suri. Stochastic minimum spanning trees in euclidean spaces. In Proceedings of the 27th Symposium on Computational Geometry, pages 65-74, 2011.
H. Kruger. Basic measures for imprecise point sets in ℝ^d. Master’s thesis, Utrecht University, 2008.
M. Langberg and L.J. Schulman. Universal ε-approximators for integrals. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, 2010.
J. Li and H. Wang. Range queries on uncertain data. Theoretical Computer Science, 609(1):32-48, 2016.
M. Löffler and J. Phillips. Shape fitting on point sets with probability distributions. In Proceedings of the 17th European Symposium on Algorithms, pages 313-324, 2009.
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http://dx.doi.org/10.1145/1377676.1377727
M. Löffler and M. van Kreveld. Approximating largest convex hulls for imprecise points. Journal of Discrete Algorithms, 6:583-594, 2008.
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Every Property Is Testable on a Natural Class of Scale-Free Multigraphs
In this paper, we introduce a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs, and consider constant-time testability on the class. We show that a very wide subclass of HSF is hyperfinite. Based on this result, an algorithm for a deterministic partitioning oracle can be constructed. We conclude by showing that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes findings by Newman and Sohler of STOC'11. However, their algorithm is based on a bounded-degree model, while it is known that actual scale-free networks usually include hubs, which have a very large degree. HSF is based on scale-free properties and includes such hubs. This is the first universal result of constant-time testability on a class of graphs made by a model of scale-free networks, and it has the potential to be applicable on a very wide range of scale-free networks.
constant-time algorithms
scale-free networks
complex networks
isolated cliques
hyperfinite
51:1-51:12
Regular Paper
Hiro
Ito
Hiro Ito
10.4230/LIPIcs.ESA.2016.51
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Explicit Correlation Amplifiers for Finding Outlier Correlations in Deterministic Subquadratic Time
We derandomize G. Valiant's [J.ACM 62(2015) Art.13] subquadratic-time algorithm for finding outlier correlations in binary data. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant's randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders in Reingold, Vadhan, and Wigderson [Ann. of Math 155(2002), 157-187]. We say that a function f:{-1,1}^d ->{-1,1}^D is a correlation amplifier with threshold 0 <= tau <= 1, error gamma >= 1, and strength p an even positive integer if for all pairs of vectors x,y in {-1,1}^d it holds that (i) |<x,y>|<tau d implies |<f(x),f(y)>| <= (tau*gamma)^p*D; and (ii) |<x,y>| >= tau*d implies (<x,y>/gamma^d})^p*D <= <f(x),f(y)> <= (gamma*<x,y>/d)^p*D.
correlation
derandomization
outlier
similarity search
expander graph
52:1-52:17
Regular Paper
Matti
Karppa
Matti Karppa
Petteri
Kaski
Petteri Kaski
Jukka
Kohonen
Jukka Kohonen
Padraig
Ó Catháin
Padraig Ó Catháin
10.4230/LIPIcs.ESA.2016.52
Thomas D. Ahle, Rasmus Pagh, Ilya Razenshteyn, and Francesco Silvestri. On the complexity of inner product similarity join. arXiv, abs/1510.02824, 2015.
Josh Alman and Ryan Williams. Probabilistic polynomials and Hamming nearest neighbors. In Proc. 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 136-150, Los Alamitos, CA, USA, 2015. IEEE Computer Society.
Noga Alon. Problems and results in extremal combinatorics - I. Discrete Math., 273(1-3):31-53, 2003.
Alexandr Andoni, Piotr Indyk, Huy L. Nguyen, and Ilya Razenshteyn. Beyond locality-sensitive hashing. In Proc. 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1018-1028, Philadelphia, PA, USA, 2014. Society for Industrial and Applied Mathematics. URL: http://dx.doi.org/10.1137/1.9781611973402.76.
http://dx.doi.org/10.1137/1.9781611973402.76
Alexandr Andoni and Ilya Razenshteyn. Optimal data-dependent hashing for approximate near neighbors. In Proc. 47th ACM Annual Symposium on the Theory of Computing (STOC), pages 793-801, New York, NY, USA, 2015. Association for Computing Machinery. URL: http://dx.doi.org/10.1145/2746539.2746553.
http://dx.doi.org/10.1145/2746539.2746553
L. Elisa Celis, Omer Reingold, Gil Segev, and Udi Wieder. Balls and bins: Smaller hash families and faster evaluation. SIAM J. Comput., 42(3):1030-1050, 2013.
Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Robert Krauthgamer, editor, Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1246-1255, Arlington, VA, USA, 2016. Society for Industrial and Applied Mathematics.
Moshe Dubiner. Bucketing coding and information theory for the statistical high-dimensional nearest-neighbor problem. IEEE Trans. Inf. Theory, 56(8):4166-4179, 2010. URL: http://dx.doi.org/10.1109/TIT.2010.2050814.
http://dx.doi.org/10.1109/TIT.2010.2050814
Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. On agnostic learning of parities, monomials, and halfspaces. SIAM J. Comput., 39(2):606-645, 2009. URL: http://dx.doi.org/10.1137/070684914.
http://dx.doi.org/10.1137/070684914
Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Similarity search in high dimensions via hashing. In Malcolm P. Atkinson, Maria E. Orlowska, Patrick Valduriez, Stanley B. Zdonik, and Michael L. Brodie, editors, Proc. 25th International Conference on Very Large Data Bases (VLDB'99), pages 518-529, Edinburgh, Scotland, UK, 1999. Morgan Kaufmann.
Parikshit Gopalan, Daniek Kane, and Raghu Meka. Pseudorandomness via the Discrete Fourier Transform. In Proc. IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 903-922, Berkeley, CA, USA, 2015. IEEE Computer Society.
Parikshit Gopalan, Raghu Meka, Omer Reingold, and David Zuckerman. Pseudorandom generators for combinatorial shapes. SIAM J. Comput., 42(3):1051-1076, 2013.
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http://dx.doi.org/10.1007/978-3-642-24412-4_32
Wassily Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58:13-30, 1963.
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http://dx.doi.org/10.1006/jcss.2000.1727
Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proc. 30th Annual ACM Symposium on the Theory of Computing (STOC), pages 604-613, New York, NY, USA, 1998. Association for Computing Machinery. URL: http://dx.doi.org/10.1145/276698.276876.
http://dx.doi.org/10.1145/276698.276876
Daniel M. Kane, Raghu Meka, and Jelani Nelson. Almost optimal explicit Johnson-Lindenstrauss families. In Proc. 14th International Workshop on Approximation, Randomization, and Combinatorial Optimization, RANDOM and 15th International Workshop on Algorithms and Techniques, APPROX, pages 628-639, Princeton, NJ, USA, 2011.
Michael Kapralov. Smooth tradeoffs between insert and query complexity in nearest neighbor search. In Proc. 34th ACM Symposium on Principles of Database Systems (PODS), pages 329-342, New York, NY, USA, 2015. Association for Computing Machinery. URL: http://dx.doi.org/10.1145/2745754.2745761.
http://dx.doi.org/10.1145/2745754.2745761
Matti Karppa, Petteri Kaski, and Jukka Kohonen. A faster subquadratic algorithm for finding outlier correlations. In Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA), pages 1288-1305, Arlington, VA, USA, 2016. Society for Industrial and Applied Mathematics.
Pravesh K. Kothari and Raghu Meka. Almost optimal pseudorandom generators for spherical caps. In Proc. 47th Annual ACM Symposium on Theory of Computing (STOC), pages 247-256, Portland, OR, USA, 2015.
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http://dx.doi.org/10.1109/FOCS.2012.80
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Alexander May and Ilya Ozerov. On computing nearest neighbors with applications to decoding of binary linear codes. In Proc. EUROCRYPT 2015 - 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 203-228, Berlin, Germany, 2015. Springer. URL: http://dx.doi.org/10.1007/978-3-662-46800-5_9.
http://dx.doi.org/10.1007/978-3-662-46800-5_9
Elchanan Mossel, Ryan O'Donnell, and Rocco A. Servedio. Learning functions of k relevant variables. J. Comput. Syst. Sci., 69(3):421-434, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2004.04.002.
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Rajeev Motwani, Assaf Naor, and Rina Panigrahy. Lower bounds on locality sensitive hashing. SIAM J. Discrete Math., 21(4):930-935, 2007. URL: http://dx.doi.org/10.1137/050646858.
http://dx.doi.org/10.1137/050646858
Ryan O'Donnell, Yi Wu, and Yuan Zhou. Optimal lower bounds for locality-sensitive hashing (except when q is tiny). ACM Trans. Comput. Theory, 6(1):Article 5, 2014. URL: http://dx.doi.org/10.1145/2578221.
http://dx.doi.org/10.1145/2578221
Rasmus Pagh. Locality-sensitive hashing without false negatives. In Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1-9, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics.
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Ninh Pham and Rasmus Pagh. Scalability and total recall with fast CoveringLSH. arXiv, abs/1602.02620, 2016.
Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. of Math., 155(1):157-187, 2002.
Victor Shoup. New algorithms for finding irreducible polynomials over finite fields. Math. Comp., 54:435-447, 1990.
Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and the closest pair problem. J. ACM, 62(2):Article 13, 2015. URL: http://dx.doi.org/10.1145/2728167.
http://dx.doi.org/10.1145/2728167
Leslie G. Valiant. Functionality in neural nets. In Proc. 1st Annual Workshop on Computational Learning Theory (COLT), pages 28-39, New York, NY, USA, 1988. Association for Computing Machinery.
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Faster Worst Case Deterministic Dynamic Connectivity
We present a deterministic dynamic connectivity data structure for undirected graphs with worst case update time O(sqrt{(n(log(log(n)))^2)/log(n)}) and constant query time. This improves on the previous best deterministic worst case algorithm of Frederickson (SIAM J. Comput., 1985) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM, 1997), which had update time O(sqrt{n}). All other algorithms for dynamic connectivity are either randomized (Monte Carlo) or have only amortized performance guarantees.
dynamic graph
spanning tree
53:1-53:15
Regular Paper
Casper
Kejlberg-Rasmussen
Casper Kejlberg-Rasmussen
Tsvi
Kopelowitz
Tsvi Kopelowitz
Seth
Pettie
Seth Pettie
Mikkel
Thorup
Mikkel Thorup
10.4230/LIPIcs.ESA.2016.53
U. A. Acar, G. E. Blelloch, R. Harper, J. L. Vittes, and S. L. M. Woo. Dynamizing static algorithms, with applications to dynamic trees and history independence. In Proceedings 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 531-540, 2004.
S. Albers and T. Hagerup. Improved parallel integer sorting without concurrent writing. Information and Computation, 136(1):25-51, 1997. URL: http://dx.doi.org/10.1006/inco.1997.2632.
http://dx.doi.org/10.1006/inco.1997.2632
S. Alstrup, J. Holm, K. de Lichtenberg, and M. Thorup. Maintaining information in fully dynamic trees with top trees. ACM Trans. on Algorithms, 1(2):243-264, 2005. URL: http://dx.doi.org/10.1145/1103963.1103966.
http://dx.doi.org/10.1145/1103963.1103966
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms, 3rd ed. MIT Press, 2009.
E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269-271, 1959.
D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification-a technique for speeding up dynamic graph algorithms. In Proceedings 33rd Annual Symposium on Foundations of Computer Science (FOCS), pages 60-69, 1992.
D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. J. ACM, 44(5):669-696, 1997.
D. Eppstein, G. F. Italiano, R. Tamassia, R. E. Tarjan, J. Westbrook, and M. Yung. Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algor., 13(1):33-54, 1992.
D. Eppstein, G. F. Italiano, R. Tamassia, R. E. Tarjan, J. Westbrook, and M. Yung. Corrigendum: Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algor., 15(1):173, 1993.
G. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781-798, 1985.
M. L. Fredman and M. Saks. The cell probe complexity of dynamic data structures. In Proceedings 21st Annual ACM Symposium on Theory of Computing (STOC), pages 345-354, 1989.
M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34(3):596-615, 1987.
D. Gibb, B. M. Kapron, V. King, and N. Thorn. Dynamic graph connectivity with improved worst case update time and sublinear space. CoRR, abs/1509.06464, 2015.
M. R. Henzinger and M. L. Fredman. Lower bounds for fully dynamic connectivity problems in graphs. Algorithmica, 22(3):351-362, 1998.
M. R. Henzinger and V. King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM, 46(4):502-516, 1999.
M. R. Henzinger and M. Thorup. Sampling to provide or to bound: With applications to fully dynamic graph algorithms. J. Random Structures and Algs., 11(4):369-379, 1997.
J. Holm, K. de Lichtenberg, and M. Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, 2001.
B. M. Kapron, V. King, and B. Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1131-1142, 2013.
P. B. Miltersen, S. Subramanian, J. S. Vitter, and R. Tamassia. Complexity models for incremental computation. Theoretical Computer Science, 130(1):203-236, 1994.
M. Pǎtraşcu and E. Demaine. Logarithmic lower bounds in the cell-probe model. SIAM J. Comput., 35(4):932-963, 2006.
M. Patrascu and M. Thorup. Don't rush into a union: take time to find your roots. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pages 559-568, 2011. Technical report available as arXiv:1102.1783.
D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26(3):362-391, 1983.
R. E. Tarjan and R. F. Werneck. Self-adjusting top trees. In Proceedings 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 813-822, 2005.
M. Thorup. Near-optimal fully-dynamic graph connectivity. In Proceedings 32nd ACM Symposium on Theory of Computing (STOC), pages 343-350, 2000.
C. Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1757-1769, 2013.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Think Eternally: Improved Algorithms for the Temp Secretary Problem and Extensions
The Temp Secretary Problem was recently introduced by [Fiat et al., ESA 2015]. It is a generalization of the Secretary Problem, in which commitments are temporary for a fixed duration. We present a simple online algorithm with improved performance guarantees for cases already considered by [Fiat et al., ESA 2015] and give competitive ratios for new generalizations of the problem. In the classical setting, where candidates have identical contract durations gamma << 1 and we are allowed to hire up to B candidates simultaneously, our algorithm is (1/2) - O(sqrt{gamma})-competitive. For large B, the bound improves to 1 - O(1/sqrt{B}) - O(sqrt{gamma}).
Furthermore we generalize the problem from cardinality constraints towards general packing constraints. We achieve a competitive ratio of 1 - O(sqrt{(1+log(d) + log(B))/B}) - O(sqrt{gamma}), where d is the sparsity of the constraint matrix and B is generalized to the capacity ratio of linear constraints. Additionally we extend the problem towards arbitrary hiring durations.
Our algorithmic approach is a relaxation that aggregates all temporal constraints into a non-temporal constraint. Then we apply a linear scaling algorithm that, on every arrival, computes a tentative solution on the input that is known up to this point. This tentative solution uses the non-temporal, relaxed constraints scaled down linearly by the amount of time that has already passed.
Secretary Problem
Online Algorithms
Scheduling Problems
54:1-54:17
Regular Paper
Thomas
Kesselheim
Thomas Kesselheim
Andreas
Tönnis
Andreas Tönnis
10.4230/LIPIcs.ESA.2016.54
Shipra Agrawal and Nikhil R. Devanur. Fast algorithms for online stochastic convex programming. In Proc. 26th Symp. Discr. Algorithms (SODA), pages 1405-1424, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.93.
http://dx.doi.org/10.1137/1.9781611973730.93
Shipra Agrawal, Zizhuo Wang, and Yinyu Ye. A dynamic near-optimal algorithm for online linear programming. Operations Research, 62(4):876-890, 2014. URL: http://dx.doi.org/10.1287/opre.2014.1289.
http://dx.doi.org/10.1287/opre.2014.1289
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. A knapsack secretary problem with applications. In Proc. 10thIntl. Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 16-28, 2007. URL: http://dx.doi.org/10.1007/978-3-540-74208-1_2.
http://dx.doi.org/10.1007/978-3-540-74208-1_2
Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Proc. 18th Symp. Discr. Algorithms (SODA), pages 434-443, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283429.
http://dl.acm.org/citation.cfm?id=1283383.1283429
MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Morteza Zadimoghaddam. Submodular secretary problem and extensions. ACM Trans. Algorithms, 9(4):32, 2013. URL: http://dx.doi.org/10.1145/2500121.
http://dx.doi.org/10.1145/2500121
Nikhil R. Devanur, Kamal Jain, Balasubramanian Sivan, and Christopher A. Wilkens. Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In Proc. 12thConf. Econom. Comput. (EC), pages 29-38, 2011. URL: http://dx.doi.org/10.1145/1993574.1993581.
http://dx.doi.org/10.1145/1993574.1993581
Eugene B. Dynkin. The optimum choice of the instant for stopping a markov process. In Sov. Math. Dokl, volume 4, pages 627-629, 1963.
Hossein Esfandiari, Nitish Korula, and Vahab S. Mirrokni. Online allocation with traffic spikes: Mixing adversarial and stochastic models. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC'15, Portland, OR, USA, June 15-19, 2015, pages 169-186, 2015. URL: http://dx.doi.org/10.1145/2764468.2764536.
http://dx.doi.org/10.1145/2764468.2764536
Moran Feldman, Joseph Naor, and Roy Schwartz. Improved competitive ratios for submodular secretary problems (extended abstract). In Proc. 14thIntl. Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 218-229, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22935-0_19.
http://dx.doi.org/10.1007/978-3-642-22935-0_19
Moran Feldman, Ola Svensson, and Rico Zenklusen. A simple O(log log(rank))-competitive algorithm for the matroid secretary problem. In Proc. 26th Symp. Discr. Algorithms (SODA), pages 1189-1201, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.79.
http://dx.doi.org/10.1137/1.9781611973730.79
Amos Fiat, Ilia Gorelik, Haim Kaplan, and Slava Novgorodov. The temp secretary problem. In Proc. 23rdEuropean Symp. Algorithms (ESA), pages 631-642, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_53.
http://dx.doi.org/10.1007/978-3-662-48350-3_53
M. Gardner. Scientific american, 1960.
Anupam Gupta and Marco Molinaro. How experts can solve LPs online. In Proc. 22ndEuropean Symp. Algorithms (ESA), pages 517-529, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_43.
http://dx.doi.org/10.1007/978-3-662-44777-2_43
Edward G. Coffman Jr., Philippe Flajolet, Leopold Flatto, and Micha Hofri. The maximum of a random walk and its application to rectangle packing. Probability in the Engineering and Informational Sciences, 12(3):373-386, 1998. URL: http://dx.doi.org/10.1017/S0269964800005258.
http://dx.doi.org/10.1017/S0269964800005258
Thomas Kesselheim, Robert D. Kleinberg, and Rad Niazadeh. Secretary problems with non-uniform arrival order. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 879-888, 2015. URL: http://dx.doi.org/10.1145/2746539.2746602.
http://dx.doi.org/10.1145/2746539.2746602
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An optimal online algorithm for weighted bipartite matching and extensions to combinatorial auctions. In Proc. 21st European Symp. Algorithms (ESA), pages 589-600, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_50.
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http://dx.doi.org/10.1016/0304-3975(94)90150-3
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Hardness of Bipartite Expansion
We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U,V,E) and a parameter beta, the goal is to find a subset V' subseteq V containing beta fraction of the vertices of V which minimizes the size of N(V'), the neighborhood of V'. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion.
In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0
there is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether
- (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or
- (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless
NP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms.
We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of
Raghavendra and Steurer 2010.
inapproximability
bipartite expansion
PCP
submodular minimization
55:1-55:17
Regular Paper
Subhash
Khot
Subhash Khot
Rishi
Saket
Rishi Saket
10.4230/LIPIcs.ESA.2016.55
A. Agarwal, M. Charikar, K. Makarychev, and Y. Makarychev. O(sqrt(log n)) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In Proceedings of the ACM Symposium on the Theory of Computing, pages 573-581, 2005.
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Creative Commons Attribution 3.0 Unported license
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A Streaming Algorithm for the Undirected Longest Path Problem
We present the first streaming algorithm for the longest path problem in undirected graphs. The input graph is given as a stream of edges and RAM is limited to only a linear number of edges at a time (linear in the number of vertices n). We prove a per-edge processing time of O(n), where a naive solution would have required Omega(n^2). Moreover, we give a concrete linear upper bound on the number of bits of RAM that are required.
On a set of graphs with various structure, we experimentally compare our algorithm with three leading RAM algorithms: Warnsdorf (1823), Pohl-Warnsdorf (1967), and Pongrasz (2012). Although conducting only a small constant number of passes over the input, our algorithm delivers competitive results: with the exception of preferential attachment graphs, we deliver at least 71% of the solution of the best RAM algorithm. The same minimum relative performance of 71% is observed over all graph classes after removing the 10% worst cases. This comparison has strong meaning, since for each instance class there is one algorithm that on average delivers at least 84% of a Hamilton path. In some cases we deliver even better results than any of the RAM algorithms.
Streaming Algorithms
Undirected Longest Path Problem
Graph Algorithms
Combinatorial Optimization
56:1-56:17
Regular Paper
Lasse
Kliemann
Lasse Kliemann
Christian
Schielke
Christian Schielke
Anand
Srivastav
Anand Srivastav
10.4230/LIPIcs.ESA.2016.56
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http://dx.doi.org/10.1016/j.ipl.2008.11.004
Creative Commons Attribution 3.0 Unported license
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A Note On Spectral Clustering
Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering.
Peng et al. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing.
We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result.
spectral embedding
k-means clustering
power method
gap assumption
57:1-57:14
Regular Paper
Pavel
Kolev
Pavel Kolev
Kurt
Mehlhorn
Kurt Mehlhorn
10.4230/LIPIcs.ESA.2016.57
Christos Boutsidis, Prabhanjan Kambadur, and Alex Gittens. Spectral clustering via the power method - provably. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 40-48, 2015.
Christos Boutsidis and Malik Magdon-Ismail. Faster svd-truncated regularized least-squares. In 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29 - July 4, 2014, pages 1321-1325, 2014.
James R. Lee, Shayan Oveis Gharan, and Luca Trevisan. Multi-way spectral partitioning and higher-order Cheeger inequalities. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 1117-1130, New York, NY, USA, 2012. ACM.
Andrew Y. Ng, Michael I. Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems 14, pages 849-856. MIT Press, 2002.
Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, and Chaitanya Swamy. The effectiveness of Lloyd-type methods for the k-means problem. J. ACM, 59(6):28:1-28:22, January 2013.
Shayan Oveis Gharan and Luca Trevisan. Partitioning into expanders. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1256-1266, 2014.
Richard Peng, He Sun, and Luca Zanetti. Partitioning well-clustered graphs: Spectral clustering works! In Proceedings of The 28th Conference on Learning Theory, COLT 2015, Paris, France, July 3-6, 2015, pages 1423-1455, 2015.
Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888-905, 2000.
Ali Kemal Sinop. How to round subspaces: A new spectral clustering algorithm. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1832-1847, 2016.
Ulrike von Luxburg. A tutorial on spectral clustering. Statistics and Computing, pages 395-416, 2007.
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On the Fine-Grained Complexity of Rainbow Coloring
The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.
graph coloring
computational complexity
lower bounds
exponential time hypothesis
FPT algorithms
58:1-58:16
Regular Paper
Lukasz
Kowalik
Lukasz Kowalik
Juho
Lauri
Juho Lauri
Arkadiusz
Socala
Arkadiusz Socala
10.4230/LIPIcs.ESA.2016.58
Prabhanjan Ananth, Meghana Nasre, and Kanthi K. Sarpatwar. Rainbow connectivity: Hardness and tractability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011), pages 241-251, 2011.
Yair Caro, Arie Lev, Yehuda Roditty, Zsolt Tuza, and Raphael Yuster. On rainbow connection. Electron. J. Combin, 15(1):R57, 2008.
Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and algorithms for rainbow connection. J. of Combinatorial Optimization, 21(3):330-347, 2009.
L. Sunil Chandran and Deepak Rajendraprasad. Rainbow Colouring of Split and Threshold Graphs. Computing and Combinatorics, pages 181-192, 2012.
L. Sunil Chandran and Deepak Rajendraprasad. Inapproximability of rainbow colouring. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013), pages 153-162, 2013.
L. Sunil Chandran, Deepak Rajendraprasad, and Marek Tesař. Rainbow colouring of split graphs. Discrete Applied Mathematics, 2015. URL: http://dx.doi.org/10.1016/j.dam.2015.05.021.
http://dx.doi.org/10.1016/j.dam.2015.05.021
Gary Chartrand, Garry L. Johns, Kathleen A. McKeon, and Ping Zhang. Rainbow connection in graphs. Mathematica Bohemica, 133(1), 2008.
Gary Chartrand and Ping Zhang. Chromatic graph theory. CRC press, 2008.
Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socała. Tight bounds for graph homomorphism and subgraph isomorphism. In Proc. of the 27th Annual ACM-SIAM Symp. on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1643-1649, 2016.
Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Dániel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Reinhard Diestel. Graph Theory. Springer-Verlag Heidelberg, 2010.
Eduard Eiben, Robert Ganian, and Juho Lauri. On the complexity of rainbow coloring problems. In Proceedings of the Twenty-Sixth International Workshop on Combinatorial Algorithms, IWOCA 2015, Verona, Italy, October 5-7, pages 209-220, 2015. URL: http://arxiv.org/abs/1510.03614, URL: http://dx.doi.org/10.1007/978-3-319-29516-9_18.
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Michael Held and Richard M. Karp. A dynamic programming approach to sequencing problems. Journal of SIAM, 10:196-210, 1962.
Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
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Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Stasys Jukna. On set intersection representations of graphs. Journal of Graph Theory, 61(1):55-75, 2009. URL: http://dx.doi.org/10.1002/jgt.20367.
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A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter
In the Vertex Cover problem we are given a graph G=(V,E) and an integer k and have to determine whether there is a set X subseteq V of size at most k such that each edge in E has at least one endpoint in X. The problem can be easily solved in time O^*(2^k), making it fixed-parameter tractable (FPT) with respect to k. While the fastest known algorithm takes only time O^*(1.2738^k), much stronger improvements have been obtained by studying parameters that are smaller than k. Apart from treewidth-related results, the arguably best algorithm for Vertex Cover runs in time O^*(2.3146^p), where p = k - LP(G) is only the excess of the solution size k over the best fractional vertex cover [Lokshtanov et al., TALG 2014]. Since p <= k but k cannot be bounded in terms of p alone, this strictly increases the range of tractable instances.
Recently, Garg and Philip (SODA 2016) greatly contributed to understanding the parameterized complexity of the Vertex Cover problem. They prove that 2LP(G) - MM(G) is a lower bound for the vertex cover size of G, where MM(G) is the size of a largest matching of G, and proceed to study parameter l = k - (2LP(G)-MM(G)). They give an algorithm of running time O^*(3^l), proving that Vertex Cover is FPT in l. It can be easily observed that l <= p whereas p cannot be bounded in terms of l alone. We complement the work of Garg and Philip by proving that Vertex Cover admits a randomized polynomial kernelization in terms of l, i.e., an efficient preprocessing to size polynomial in l. This improves over parameter p = k - LP(G) for which this was previously known [Kratsch and Wahlström, FOCS 2012].
Vertex cover
parameterized complexity
kernelization
59:1-59:17
Regular Paper
Stefan
Kratsch
Stefan Kratsch
10.4230/LIPIcs.ESA.2016.59
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
http://dx.doi.org/10.1016/j.jcss.2009.04.001
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
http://dx.doi.org/10.1016/j.tcs.2011.04.039
Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex cover: Further observations and further improvements. J. Algorithms, 41(2):280-301, 2001. URL: http://dx.doi.org/10.1006/jagm.2001.1186.
http://dx.doi.org/10.1006/jagm.2001.1186
Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theor. Comput. Sci., 411(40-42):3736-3756, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.06.026.
http://dx.doi.org/10.1016/j.tcs.2010.06.026
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. On the hardness of losing width. Theory Comput. Syst., 54(1):73-82, 2014. URL: http://dx.doi.org/10.1007/s00224-013-9480-1.
http://dx.doi.org/10.1007/s00224-013-9480-1
Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. TOCT, 5(1):3, 2013. URL: http://dx.doi.org/10.1145/2462896.2462899.
http://dx.doi.org/10.1145/2462896.2462899
Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: http://dx.doi.org/10.1145/2629620.
http://dx.doi.org/10.1145/2629620
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
http://dx.doi.org/10.1007/978-1-4471-5559-1
Shivam Garg and Geevarghese Philip. Raising the bar for vertex cover: Fixed-parameter tractability above a higher guarantee. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1152-1166. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch80.
http://dx.doi.org/10.1137/1.9781611974331.ch80
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Bart M. P. Jansen. The power of data reduction: Kernels for fundamental graph problems. PhD thesis, Utrecht University, 2013.
Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst., 53(2):263-299, 2013. URL: http://dx.doi.org/10.1007/s00224-012-9393-4.
http://dx.doi.org/10.1007/s00224-012-9393-4
Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
http://eatcs.org/beatcs/index.php/beatcs/article/view/285
Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. CoRR, abs/1111.2195, 2011. URL: http://arxiv.org/abs/1111.2195.
http://arxiv.org/abs/1111.2195
Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 450-459. IEEE Computer Society, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.46.
http://dx.doi.org/10.1109/FOCS.2012.46
Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11(2):15:1-15:31, 2014. URL: http://dx.doi.org/10.1145/2566616.
http://dx.doi.org/10.1145/2566616
László Lovász and Michael D. Plummer. Matching Theory. North-Holland, 1986.
Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: Maxsat and maxcut. J. Algorithms, 31(2):335-354, 1999. URL: http://dx.doi.org/10.1006/jagm.1998.0996.
http://dx.doi.org/10.1006/jagm.1998.0996
Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, and C. R. Subramanian. The complexity of König subgraph problems and above-guarantee vertex cover. Algorithmica, 61(4):857-881, 2011. URL: http://dx.doi.org/10.1007/s00453-010-9412-2.
http://dx.doi.org/10.1007/s00453-010-9412-2
N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. LP can be a cure for parameterized problems. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 338-349. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2012.338.
http://dx.doi.org/10.4230/LIPIcs.STACS.2012.338
George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Math. Program., 8(1):232-248, 1975. URL: http://dx.doi.org/10.1007/BF01580444.
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http://dx.doi.org/10.1007/978-3-642-23719-5_33
Igor Razgon and Barry O'Sullivan. Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.002.
http://dx.doi.org/10.1016/j.jcss.2009.04.002
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The Strongly Stable Roommates Problem
An instance of the strongly stable roommates problem with incomplete lists and ties (SRTI) is an undirected non-bipartite graph G = (V,E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge {x, y} in E\M is a blocking edge for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly prefers x to its current partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We present an O(nm) time algorithm for computing a strongly stable matching, where we denote n = |V| and m = |E|. The best previously known solution had running time O(m^2) [Scott, 2005]. We also give a characterisation of the set of all strongly stable matchings. We show that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and we give an O(nm) algorithm for constructing such a representation. Our algorithms are based on a simple reduction to the bipartite version of the problem.
strongly stable matching
stable roommates
rotations
matching theory
60:1-60:15
Regular Paper
Adam
Kunysz
Adam Kunysz
10.4230/LIPIcs.ESA.2016.60
Brian C. Dean and Siddharth Munshi. Faster algorithms for stable allocation problems. Algorithmica, 58(1):59-81, 2010. URL: http://dx.doi.org/10.1007/s00453-010-9416-y.
http://dx.doi.org/10.1007/s00453-010-9416-y
Tamás Fleiner, Robert W. Irving, and David F. Manlove. Efficient algorithms for generalized stable marriage and roommates problems. Theor. Comput. Sci., 381(1-3):162-176, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.04.029.
http://dx.doi.org/10.1016/j.tcs.2007.04.029
Dan Gusfield and Robert W. Irving. The Stable marriage problem - structure and algorithms. Foundations of computing series. MIT Press, 1989.
Robert W. Irving. An efficient algorithm for the "stable roommates" problem. J. Algorithms, 6(4):577-595, 1985. URL: http://dx.doi.org/10.1016/0196-6774(85)90033-1.
http://dx.doi.org/10.1016/0196-6774(85)90033-1
Robert W. Irving. Stable marriage and indifference. Discrete Applied Mathematics, 48(3):261-272, 1994. URL: http://dx.doi.org/10.1016/0166-218X(92)00179-P.
http://dx.doi.org/10.1016/0166-218X(92)00179-P
Robert W. Irving and David F. Manlove. The stable roommates problem with ties. J. Algorithms, 43(1):85-105, 2002. URL: http://dx.doi.org/10.1006/jagm.2002.1219.
http://dx.doi.org/10.1006/jagm.2002.1219
Telikepalli Kavitha, Kurt Mehlhorn, Dimitrios Michail, and Katarzyna E. Paluch. Strongly stable matchings in time O(nm) and extension to the hospitals-residents problem. ACM Trans. Algorithms, 3(2), 2007. URL: http://dx.doi.org/10.1145/1240233.1240238.
http://dx.doi.org/10.1145/1240233.1240238
Adam Kunysz, Katarzyna E. Paluch, and Pratik Ghosal. Characterisation of strongly stable matchings. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 107-119, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch8.
http://dx.doi.org/10.1137/1.9781611974331.ch8
David F. Manlove. Stable marriage with ties and unacceptable partners. Technical report, University of Glasgow, 1999.
David F. Manlove. The structure of stable marriage with indifference. Discrete Applied Mathematics, 122(1-3):167-181, 2002. URL: http://dx.doi.org/10.1016/S0166-218X(01)00322-5.
http://dx.doi.org/10.1016/S0166-218X(01)00322-5
David F. Manlove. Algorithmics of Matching Under Preferences, volume 2 of Series on Theoretical Computer Science. WorldScientific, 2013. URL: http://dx.doi.org/10.1142/8591.
http://dx.doi.org/10.1142/8591
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Nitsan Perach, Julia Polak, and Uriel G. Rothblum. A stable matching model with an entrance criterion applied to the assignment of students to dormitories at the technion. Int. J. Game Theory, 36(3-4):519-535, 2008. URL: http://dx.doi.org/10.1007/s00182-007-0083-4.
http://dx.doi.org/10.1007/s00182-007-0083-4
Nitsan Perach and Uriel G. Rothblum. Incentive compatibility for the stable matching model with an entrance criterion. Int. J. Game Theory, 39(4):657-667, 2010. URL: http://dx.doi.org/10.1007/s00182-009-0210-5.
http://dx.doi.org/10.1007/s00182-009-0210-5
Eytan Ronn. Np-complete stable matching problems. J. Algorithms, 11(2):285-304, 1990. URL: http://dx.doi.org/10.1016/0196-6774(90)90007-2.
http://dx.doi.org/10.1016/0196-6774(90)90007-2
Sandy Scott. A study of stable marriage problems with ties. PhD thesis, University of Glasgow, 2005.
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Faster External Memory LCP Array Construction
The suffix array, perhaps the most important data structure in modern string processing, needs to be augmented with the longest-common-prefix (LCP) array in many applications. Their construction is often a major bottleneck especially when the data is too big for internal memory. We describe two new algorithms for computing the LCP array from the suffix array in external memory. Experiments demonstrate that the new algorithms are about a factor of two faster than the fastest previous algorithm.
LCP array
suffix array
external memory algorithms
61:1-61:16
Regular Paper
Juha
Kärkkäinen
Juha Kärkkäinen
Dominik
Kempa
Dominik Kempa
10.4230/LIPIcs.ESA.2016.61
M. I. Abouelhoda, S. Kurtz, and E. Ohlebusch. Replacing suffix trees with enhanced suffix arrays. J. Discrete Algorithms, 2(1):53-86, 2004. URL: http://dx.doi.org/10.1016/S1570-8667(03)00065-0.
http://dx.doi.org/10.1016/S1570-8667(03)00065-0
M. J. Bauer, A. J. Cox, G. Rosone, and M. Sciortino. Lightweight LCP construction for next-generation sequencing datasets. In Proceedings of the 12th Workshop on Algorithms in Bioinformatics (WABI 2012), volume 7534 of LNCS, pages 326-337. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33122-0_26.
http://dx.doi.org/10.1007/978-3-642-33122-0_26
T. Beller, S. Gog, E. Ohlebusch, and T. Schnattinger. Computing the longest common prefix array based on the Burrows-Wheeler transform. J. Discrete Algorithms, 18:22-31, 2013. URL: http://dx.doi.org/10.1016/j.jda.2012.07.007.
http://dx.doi.org/10.1016/j.jda.2012.07.007
T. Bingmann, J. Fischer, and V. Osipov. Inducing suffix and LCP arrays in external memory. In Proceedings of the 2013 Workshop on Algorithm Engineering and Experiments (ALENEX 2013), pages 88-102. SIAM, 2013. URL: http://dx.doi.org/10.1137/1.9781611972931.8.
http://dx.doi.org/10.1137/1.9781611972931.8
D. R. Clark. Compact Pat Trees. PhD thesis, University of Waterloo, 1998.
F. A. da Louza, G. P. Telles, and C. D. de Aguiar Ciferri. External memory generalized suffix and LCP arrays construction. In Proceedings of the 24th Annual Symposium on Combinatorial Pattern Matching (CPM 2013), volume 7922 of LNCS, pages 201-210. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-38905-4_20.
http://dx.doi.org/10.1007/978-3-642-38905-4_20
R. Dementiev, J. Kärkkäinen, J. Mehnert, and P. Sanders. Better external memory suffix array construction. ACM J. Exp. Algor., 12:3.4:1-3.4:24, August 2008. URL: http://dx.doi.org/10.1145/1227161.1402296.
http://dx.doi.org/10.1145/1227161.1402296
M. Deo and S. Keely. Parallel suffix array and least common prefix for the GPU. In Proceedings of the 18th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPoPP 2013), pages 197-206. ACM, 2013. URL: http://dx.doi.org/10.1145/2442516.2442536.
http://dx.doi.org/10.1145/2442516.2442536
S. Gog and E. Ohlebusch. Fast and lightweight LCP-array construction algorithms. In Proceedings of the 2011 Workshop on Algorithm Engineering and Experiments (ALENEX 2011), pages 25-34. SIAM, 2011. URL: http://dx.doi.org/10.1137/1.9781611972917.3.
http://dx.doi.org/10.1137/1.9781611972917.3
G. H. Gonnet, R. A. Baeza-Yates, and T. Snider. New indices for text: Pat trees and Pat arrays. In W. B. Frakes and R. Baeza-Yates, editors, Information Retrieval: Data Structures & Algorithms, pages 66-82. Prentice-Hall, 1992.
J. Kärkkäinen and D. Kempa. Engineering a lightweight external memory suffix array construction algorithm. In Proceedings of the 2nd International Conference on Algorithms for Big Data (ICABD 2014), volume 1146 of CEUR Workshop Proceedings, pages 53-60. CEUR-WS.org, 2014.
J. Kärkkäinen and D. Kempa. LCP array construction in external memory. In Proceedings of the 13th International Symposium on Experimental Algorithms (SEA 2014), volume 8504 of LNCS, pages 412-423. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07959-2_35.
http://dx.doi.org/10.1007/978-3-319-07959-2_35
J. Kärkkäinen and D. Kempa. LCP array construction in external memory. J. Exp. Algorithmics, 21(1):1.7:1-1.7:22, April 2016. URL: http://dx.doi.org/10.1145/2851491.
http://dx.doi.org/10.1145/2851491
J. Kärkkäinen, D. Kempa, and M. Pia̧tkowski. Tighter bounds for the sum of irreducible LCP values. In Proceedings of the 26th Annual Symposium on Combinatorial Pattern Matching (CPM 2015), volume 9133 of LNCS, pages 316-328. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-19929-0_27.
http://dx.doi.org/10.1007/978-3-319-19929-0_27
J. Kärkkäinen, D. Kempa, and S. J. Puglisi. Parallel external memory suffix sorting. In Proceedings of the 26th Annual Symposium on Combinatorial Pattern Matching (CPM 2015), volume 9133 of LNCS, pages 329-342. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-19929-0_28.
http://dx.doi.org/10.1007/978-3-319-19929-0_28
J. Kärkkäinen, G. Manzini, and S. J. Puglisi. Permuted longest-common-prefix array. In Proceedings of the 20th Annual Symposium on Combinatorial Pattern Matching (CPM 2009), volume 5577 of LNCS, pages 181-192. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02441-2_17.
http://dx.doi.org/10.1007/978-3-642-02441-2_17
J. Kärkkäinen and P. Sanders. Simple linear work suffix array construction. In Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP 2003), volume 2719 of LNCS, pages 943-955. Springer, 2003. URL: http://dx.doi.org/10.1007/3-540-45061-0_73.
http://dx.doi.org/10.1007/3-540-45061-0_73
T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park. Linear-time longest-common-prefix computation in suffix arrays and its applications. In Proceedings of the 12th Annual Symposium on Combinatorial Pattern Matching (CPM 2001), volume 2089 of LNCS, pages 181-192. Springer, 2001. URL: http://dx.doi.org/10.1007/3-540-48194-X_17.
http://dx.doi.org/10.1007/3-540-48194-X_17
W. Liu, G. Nong, W. H. Chan, and Y. Wu. Induced sorting suffixes in external memory with better design and less space. In Proceedings of the 22nd International Symposium on String Processing and Information Retrieval (SPIRE 2015), volume 9309 of LNCS, pages 83-94. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-23826-5_9.
http://dx.doi.org/10.1007/978-3-319-23826-5_9
V. Mäkinen. Compact suffix array - a space efficient full-text index. Fund. Inform., 56(1-2):191-210, 2003.
V. Mäkinen, D. Belazzougui, F. Cunial, and A. I. Tomescu. Genome-Scale Algorithm Design: Biological Sequence Analysis in the Era of High-Throughput Sequencing. Cambridge University Press, 2015.
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G. Manzini. Two space saving tricks for linear time LCP array computation. In Proceedings of the 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2004), volume 3111 of LNCS, pages 372-383. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27810-8_32.
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G. Navarro and V. Mäkinen. Compressed full-text indexes. ACM Comput. Surv., 39(1):article 2, 2007. URL: http://dx.doi.org/10.1145/1216370.1216372.
http://dx.doi.org/10.1145/1216370.1216372
G. Nong, W. H. Chan, S. Q. Hu, and Y. Wu. Induced sorting suffixes in external memory. ACM Trans. Inf. Syst., 33(3), February 2015. URL: http://dx.doi.org/10.1145/2699665.
http://dx.doi.org/10.1145/2699665
G. Nong, W. H. Chan, S. Zhang, and X. F. Guan. Suffix array construction in external memory using d-critical substrings. ACM Trans. Inf. Syst., 32(1), January 2014. URL: http://dx.doi.org/10.1145/2518175.
http://dx.doi.org/10.1145/2518175
E. Ohlebusch. Bioinformatics Algorithms: Sequence Analysis, Genome Rearrangements, and Phylogenetic Reconstruction. Oldenbusch Verlag, 2013.
D. Okanohara and K. Sadakane. Practical entropy-compressed rank/select dictionary. In Proceedings of the 2007 Workshop on Algorithm Engineering and Experiments (ALENEX 2007). SIAM, 2007. URL: http://dx.doi.org/10.1137/1.9781611972870.6.
http://dx.doi.org/10.1137/1.9781611972870.6
S. J. Puglisi and A. Turpin. Space-time tradeoffs for longest-common-prefix array computation. In Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), volume 5369 of LNCS, pages 124-135. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-92182-0_14.
http://dx.doi.org/10.1007/978-3-540-92182-0_14
K. Sadakane. Succinct representations of lcp information and improvements in the compressed suffix arrays. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002), pages 225-232. ACM/SIAM, 2002.
J. Shun. Fast parallel computation of longest common prefixes. In Proceedings of the 2014 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis (SC 2014), pages 387-398. IEEE, 2014. URL: http://dx.doi.org/10.1109/SC.2014.37.
http://dx.doi.org/10.1109/SC.2014.37
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http://dx.doi.org/10.1561/0400000014
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Almost All Even Yao-Yao Graphs Are Spanners
It is an open problem whether Yao-Yao graphs YY_{k} (also known as sparse-Yao graphs) are all spanners when the integer parameter k is large enough. In this paper we show that, for any integer k >= 42, the Yao-Yao graph YY_{2k} is a t_k-spanner, with stretch factor t_k = 6.03+O(k^{-1}) when k tends to infinity. Our result generalizes the best known result which asserts that all YY_{6k} are spanners for k >= 6 [Bauer and Damian, SODA'13]. Our proof is also somewhat simpler.
Yao-Yao graph
geometric spanner
curved trapezoid
62:1-62:13
Regular Paper
Jian
Li
Jian Li
Wei
Zhan
Wei Zhan
10.4230/LIPIcs.ESA.2016.62
Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Mirela Damian, Rolf Fagerberg, André van Renssen, Perouz Taslakian, and Sander Verdonschot. Continuous Yao graphs. In Proceedings of the 26th Canadian Conference on Computational Geometry, CCCG 2014, Halifax, Nova Scotia, Canada, 2014, 2014.
Luis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O'Rourke, André van Renssen, Perouz Taslakian, Sander Verdonschot, and Ge Xia. New and improved spanning ratios for Yao graphs. JoCG, 6(2):19-53, 2015.
Matthew Bauer and Mirela Damian. An infinite class of sparse-Yao spanners. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 184-196. SIAM, 2013.
Prosenjit Bose, Paz Carmi, Sebastien Collette, and Michiel Smid. On the stretch factor of convex delaunay graphs. Journal of computational geometry, 1(1):41-56, 2010.
Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'rourke, Ben Seamone, Michiel Smid, and Stefanie Wuhrer. π/2-angle Yao graphs are spanners. International Journal of Computational Geometry &Applications, 22(01):61-82, 2012.
Prosenjit Bose, Joachim Gudmundsson, and Michiel H. M. Smid. Constructing plane spanners of bounded degree and low weight. Algorithmica, 42(3-4):249-264, 2005.
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Artur Czumaj and Hairong Zhao. Fault-tolerant geometric spanners. Discrete &Computational Geometry, 32(2):207-230, 2004.
Mirela Damian. A simple Yao-Yao-based spanner of bounded degree. arXiv preprint arXiv:0802.4325, 2008.
Mirela Damian, Nawar Molla, and Val Pinciu. Spanner properties of π/2-angle Yao graphs. In Proc. of the 25th European Workshop on Computational Geometry, pages 21-24. Citeseer, 2009.
Mirela Damian and Kristin Raudonis. Yao graphs span theta graphs. In Combinatorial Optimization and Applications, pages 181-194. Springer, 2010.
David P Dobkin, Steven J Friedman, and Kenneth J Supowit. Delaunay graphs are almost as good as complete graphs. In Foundations of Computer Science, 1987., 28th Annual Symposium on, pages 20-26. IEEE, 1987.
Nawar M El Molla. Yao spanners for wireless ad hoc networks. Master’s thesis, Villanova University, 2009.
Lujun Jia, Rajmohan Rajaraman, and Christian Scheideler. On local algorithms for topology control and routing in ad hoc networks. In Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures, pages 220-229. ACM, 2003.
Iyad A Kanj and Ge Xia. On certain geometric properties of the Yao-Yao graphs. In Combinatorial Optimization and Applications, pages 223-233. Springer, 2012.
Xiang-Yang Li. Wireless Ad Hoc and Sensor Networks: Theory and Applications. Cambridge, 6 2008.
Xiang-Yang Li, Peng-Jun Wan, and Yu Wang. Power efficient and sparse spanner for wireless ad hoc networks. In Computer Communications and Networks, 2001. Proceedings. Tenth International Conference on, pages 564-567. IEEE, 2001.
Xiang-Yang Li, Peng-Jun Wan, Yu Wang, and Ophir Frieder. Sparse power efficient topology for wireless networks. In System Sciences, 2002. HICSS. Proceedings of the 35th Annual Hawaii International Conference on, pages 3839-3848. IEEE, 2002.
Xiang-Yang Li and Yu Wang. Efficient construction of low weight bounded degree planar spanner. In Computing and Combinatorics, pages 374-384. Springer, 2003.
Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007.
David Peleg and Alejandro A Schäffer. Graph spanners. Journal of graph theory, 13(1):99-116, 1989.
Christian Schindelhauer, Klaus Volbert, and Martin Ziegler. Geometric spanners with applications in wireless networks. Computational Geometry, 36(3):197-214, 2007.
Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982.
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Online Non-Preemptive Scheduling in a Resource Augmentation Model Based on Duality
Resource augmentation is a well-established model for analyzing algorithms, particularly in the online setting. It has been successfully used for providing theoretical evidence for several heuristics in scheduling with good performance in practice. According to this model, the algorithm is applied to a more powerful environment than that of the adversary. Several types of resource augmentation for scheduling problems have been proposed up to now, including speed augmentation, machine augmentation and more recently rejection. In this paper, we present a framework that unifies the various types of resource augmentation. Moreover, it allows generalize the notion of resource augmentation for other types of resources. Our framework is based on mathematical programming and it consists of extending the domain of feasible solutions for the algorithm with respect to the domain of the adversary. This, in turn allows the natural concept of duality for mathematical programming to be used as a tool for the analysis of the algorithm's performance. As an illustration of the above ideas, we apply this framework and we propose a primal-dual algorithm for the online scheduling problem of minimizing the total weighted flow time of jobs on unrelated machines when the preemption of jobs is not allowed. This is a well representative problem for which no online algorithm with performance guarantee is known. Specifically, a strong lower bound of Omega(sqrt{n}) exists even for the offline unweighted version of the problem on a single machine. In this paper, we first show a strong negative result even when speed augmentation is used in the online setting. Then, using the generalized framework for resource augmentation and by combining speed augmentation and rejection, we present an (1+epsilon_s)-speed O(1/(epsilon_s epsilon_r))-competitive algorithm if we are allowed to reject jobs whose total weight is an epsilon_r-fraction of the weights of all jobs, for any epsilon_s > 0 and epsilon_r in (0,1). Furthermore, we extend the idea for analysis of the above problem and we propose an (1+\epsilon_s)-speed epsilon_r-rejection O({k^{(k+3)/k}}/{epsilon_{r}^{1/k}*epsilon_{s}^{(k+2)/k}})-competitive algorithm for the more general objective of minimizing the weighted l_k-norm of the flow times of jobs.
Online algorithms
Non-preemptive scheduling
Resource augmentation
Primal-dual
63:1-63:17
Regular Paper
Giorgio
Lucarelli
Giorgio Lucarelli
Nguyen
Kim Thang
Nguyen Kim Thang
Abhinav
Srivastav
Abhinav Srivastav
Denis
Trystram
Denis Trystram
10.4230/LIPIcs.ESA.2016.63
Susanne Albers, Lene M. Favrholdt, and Oliver Giel. On paging with locality of reference. J. Comput. Syst. Sci., 70(2):145-175, 2005.
S. Anand, Naveen Garg, and Amit Kumar. Resource augmentation for weighted flow-time explained by dual fitting. In Symposium on Discrete Algorithms, pages 1228-1241, 2012.
Spyros Angelopoulos, Reza Dorrigiv, and Alejandro López-Ortiz. On the separation and equivalence of paging strategies. In Proc. Symposium on Discrete Algorithms, pages 229-237, 2007.
Nikhil Bansal, Ho-Leung Chan, Rohit Khandekar, Kirk Pruhs, B Schicber, and Cliff Stein. Non-preemptive min-sum scheduling with resource augmentation. In Proc. 48th Symposium on Foundations of Computer Science, pages 614-624, 2007.
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Niv Buchbinder and Joseph Naor. The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009.
Chandra Chekuri, Sanjeev Khanna, and An Zhu. Algorithms for minimizing weighted flow time. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 84-93, 2001.
Anamitra Roy Choudhury, Syamantak Das, Naveen Garg, and Amit Kumar. Rejecting jobs to minimize load and maximum flow-time. In Proc. Symposium on Discrete Algorithms, pages 1114-1133, 2015.
Anamitra Roy Choudhury, Syamantak Das, and Amit Kumar. Minimizing weighted 𝓁_p-norm of flow-time in the rejection model. In Proc. 35th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015), volume 45, pages 25-37, 2015.
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Nikhil R. Devanur and Zhiyi Huang. Primal dual gives almost optimal energy efficient online algorithms. In Proc. 25th ACM-SIAM Symposium on Discrete Algorithms, 2014.
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Leah Epstein and Rob van Stee. Optimal on-line flow time with resource augmentation. Discrete Applied Mathematics, 154(4):611-621, 2006.
Anupam Gupta, Ravishankar Krishnaswamy, and Kirk Pruhs. Online primal-dual for non-linear optimization with applications to speed scaling. In Proc. 10th Workshop on Approximation and Online Algorithms, pages 173-186, 2012.
Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive algorithms from competitive equilibria: Non-clairvoyant scheduling under polyhedral constraints. In STOC, 2014.
Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive flow time algorithms for polyhedral scheduling. In Proc. 56th Symposium on Foundations of Computer Science, pages 506-524, 2015.
Sungjin Im, Janardhan Kulkarni, Kamesh Munagala, and Kirk Pruhs. Selfishmigrate: A scalable algorithm for non-clairvoyantly scheduling heterogeneous processors. In Proc. 55th Symposium on Foundations of Computer Science, 2014.
Sungjin Im, Shi Li, Benjamin Moseley, and Eric Torng. A dynamic programming framework for non-preemptive scheduling problems on multiple machines [extended abstract]. In Proc. 26th ACM-SIAM Symposium on Discrete Algorithms, pages 1070-1086, 2015.
Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. J. ACM, 47(4):617-643, 2000.
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Nguyen Kim Thang. Lagrangian duality in online scheduling with resource augmentation and speed scaling. In Proc. 21st European Symposium on Algorithms, pages 755-766, 2013.
David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge University Press, 2011.
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Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.
In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds.
The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants.
low-dimensional topology
triangulations of 3-manifolds
cohomology theory
Turaev-Viro invariants
combinatorial algorithms
64:1-64:16
Regular Paper
Clément
Maria
Clément Maria
Jonathan
Spreer
Jonathan Spreer
10.4230/LIPIcs.ESA.2016.64
Benjamin A. Burton. Structures of small closed non-orientable 3-manifold triangulations. J. Knot Theory Ramifications, 16(5):545-574, 2007.
Benjamin A. Burton. Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations. In Proceedings of ISSAC, pages 59-66. ACM, 2011.
Benjamin A. Burton. A new approach to crushing 3-manifold triangulations. Discrete Comput. Geom., 52(1):116-139, 2014.
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Benjamin A. Burton, Clément Maria, and Jonathan Spreer. Algorithms and complexity for Turaev-Viro invariants. In Proceedings of ICALP 2015, pages 281-293. Springer, 2015.
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http://dx.doi.org/10.1515/9783110221848
Vladimir G. Turaev and Oleg Y. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865-902, 1992.
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The Computational Complexity of Genetic Diversity
A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition, or whether a single dominant genotype emerges. Classic work by [Kalmus, J. og Genetics, 1945] has established that even in simple diploid species (species with chromosome pairs) diversity can be guaranteed as long as the heterozygous (having different alleles for a gene on two chromosomes) individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g., human blood types) predicting diversity (and its implications) is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus (gene) diploid models, predicting whether diversity survives or not given its fitness landscape is algorithmically intractable.
Our hardness results are structurally robust along several dimensions, e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes. Technically, our results exploit connections between game theory, nonlinear dynamical systems, and complexity theory and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games; finding one Nash equilibrium is easy in these games. In the process we characterize stable fixed points of these dynamics using the notions of Nash equilibrium and negative semidefiniteness. This as well as hardness results for decision problems in coordination games may be of independent interest. Finally, we complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. The full version of this paper is available at http://arxiv.org/abs/1411.6322.
Dynamical Systems
Stability
Complexity
Optimization
Equilibrium
65:1-65:17
Regular Paper
Ruta
Mehta
Ruta Mehta
Ioannis
Panageas
Ioannis Panageas
Georgios
Piliouras
Georgios Piliouras
Sadra
Yazdanbod
Sadra Yazdanbod
10.4230/LIPIcs.ESA.2016.65
E. Chastain, A. Livnat, C. H. Papadimitriou, and U. Vazirani. Algorithms, games, and evolution. PNAS, 2014. URL: http://dx.doi.org/10.1073/pnas.1406556111.
http://dx.doi.org/10.1073/pnas.1406556111
E. Chastain, A. Livnat, C. H. Papadimitriou, and U. V. Vazirani. Multiplicative updates in coordination games and the theory of evolution. In ITCS, pages 57-58, 2013. URL: http://dx.doi.org/10.1145/2422436.2422444.
http://dx.doi.org/10.1145/2422436.2422444
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Approximation and Hardness of Token Swapping
Given a graph G=(V,E) with V={1,...,n}, we place on every vertex a token T_1,...,T_n. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token T_i is on vertex i. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out any 2^{o(n)} algorithm under the ETH. This is matched with a simple 2^{O(n*log(n))} algorithm based on a breadth-first search in an auxiliary graph. We show one general 4-approximation and show APX-hardness. Thus, there is a small constant delta > 1 such that every polynomial time approximation algorithm has approximation factor at least delta.
Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.
token swapping
minimum generator sequence
graph theory
NP-hardness
approximation algorithms
66:1-66:15
Regular Paper
Tillmann
Miltzow
Tillmann Miltzow
Lothar
Narins
Lothar Narins
Yoshio
Okamoto
Yoshio Okamoto
Günter
Rote
Günter Rote
Antonis
Thomas
Antonis Thomas
Takeaki
Uno
Takeaki Uno
10.4230/LIPIcs.ESA.2016.66
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Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Linear-time algorithm for sliding tokens on trees. Theoretical Computer Science, 600:132-142, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.07.037.
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Ruy Fabila-Monroy, David Flores-Peñaloza, Clemens Huemer, Ferran Hurtado, Jorge Urrutia, and David R. Wood. Token graphs. Graphs and Combinatorics, 28:365-380, 2012. URL: http://dx.doi.org/10.1007/s00373-011-1055-9.
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Eli Fox-Epstein, Duc A. Hoang, Yota Otachi, and Ryuhei Uehara. Sliding token on bipartite permutation graphs. In Khaled Elbassioni and Kazuhisa Makino, editors, Algorithms and Computation, volume 9472 of Lecture Notes in Computer Science, pages 237-247. Springer Berlin Heidelberg, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48971-0_21.
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Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science, 343(1-2):72-96, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.05.008.
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Lenwood S. Heath and John Paul C. Vergara. Sorting by short swaps. Journal of Computational Biology, 10(5):775-789, 2003. URL: http://dx.doi.org/10.1089/106652703322539097.
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Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
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http://dx.doi.org/10.1016/j.tcs.2012.03.004
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Tillmann Miltzow, Lothar Narins, Yoshio Okamoto, Günter Rote, Antonis Thomas, and Takeaki Uno. Approximation and Hardness of Token Swapping. Preprint, February 2016. URL: http://arxiv.org/abs/1602.05150.
http://arxiv.org/abs/1602.05150
Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, and Marcin Wrochna. Reconfiguration over tree decompositions. In Marek Cygan and Pinar Heggernes, editors, Parameterized and Exact Computation, volume 8894 of Lecture Notes in Computer Science, pages 246-257. Springer International Publishing, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13524-3_21.
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Katsuhisa Yamanaka, Erik D. Demaine, Takehiro Ito, Jun Kawahara, Masashi Kiyomi, Yoshio Okamoto, Toshiki Saitoh, Akira Suzuki, Kei Uchizawa, and Takeaki Uno. Swapping labeled tokens on graphs. Theoretical Computer Science, 586:81-94, 2015. Special issue for the conference Fun with Algorithms 2014. URL: http://dx.doi.org/10.1016/j.tcs.2015.01.052.
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Katsuhisa Yamanaka, Takashi Horiyama, David Kirkpatrick, Yota Otachi, Toshiki Saitoh, Ryuhei Uehara, and Yushi Uno. Swapping colored tokens on graphs. In Frank Dehne, Jörg-Rüdiger Sack, and Ulrike Stege, editors, Algorithms and Data Structures, volume 9214 of Lecture Notes in Computer Science, pages 619-628. Springer International Publishing, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21840-3_51.
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Gaku Yasui, Kouta Abe, Katsuhisa Yamanaka, and Takashi Hirayama. Swapping labeled tokens on complete split graphs. SIG Technical Reports, 2015-AL-153(14):1-4, 2015.
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A 7/3-Approximation for Feedback Vertex Sets in Tournaments
We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001].
Approximation algorithms
feedback vertex sets
tournaments
67:1-67:14
Regular Paper
Matthias
Mnich
Matthias Mnich
Virginia
Vassilevska Williams
Virginia Vassilevska Williams
László A.
Végh
László A. Végh
10.4230/LIPIcs.ESA.2016.67
Jeffrey S. Banks. Sophisticated voting outcomes and agenda control. Soc. Choice Welf., 1(4):295-306, 1985.
Reuven Bar-Yehuda and Dror Rawitz. On the equivalence between the primal-dual schema and the local ratio technique. SIAM J. Discrete Math., 19(3):762-797, 2005.
Eli Berger, Krzysztof Choromanski, Maria Chudnovsky, Jacob Fox, Martin Loebl, Alex Scott, Paul Seymour, and Stéphan Thomassé. Tournaments and colouring. J. Combin. Theory Ser. B, 103(1):1-20, 2013.
Mao-Cheng Cai, Xiaotie Deng, and Wenan Zang. An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput., 30(6):1993-2007 (electronic), 2001.
Mao-cheng Cai, Xiaotie Deng, and Wenan Zang. A min-max theorem on feedback vertex sets. Math. Oper. Res., 27(2):361-371, 2002.
Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Ann. of Math. (2), 162(1):439-485, 2005.
Michael Dom, Jiong Guo, Falk Hüffner, Rolf Niedermeier, and Anke Truss. Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms, 8(1):76-86, 2010.
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Samuel Fiorini, Gwenaël Joret, and Oliver Schaudt. Improved approximation algorithms for hitting 3-vertex paths. In Proc. IPCO 2016, volume 9682 of Lecture Notes Comput. Sci., pages 238-249, 2016.
Fedor V. Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. In Proc. STOC 2016, pages 764-775, 2016.
Serge Gaspers and Matthias Mnich. Feedback vertex sets in tournaments. J. Graph Theory, 72(1):72-89, 2013.
Kamal Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21(1):39-60, 2001.
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Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. System Sci., 74(3):335-349, 2008.
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Mithilesh Kumar and Daniel Lokshtanov. Faster exact and parameterized algorithm for feedback vertex set in tournaments. In Proc. STACS 2016, volume 47 of Leibniz Int. Proc. Informatics, pages 49:1-49:13, 2016.
Lap Chi Lau, R. Ravi, and Mohit Singh. Iterative methods in combinatorial optimization. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York, 2011.
John W. Moon. On maximal transitive subtournaments. Proc. Edinburgh Math. Soc. (2), 17:345-349, 1970/71.
Adolfo Sanchez-Flores. On tournaments free of large transitive subtournaments. Graphs Comb., 14(2):181-200, 1998.
Prashant Sasatte. Improved approximation algorithm for the feedback set problem in a bipartite tournament. Oper. Res. Lett., 36(5):602-604, 2008.
Paul D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995.
Ewald Speckenmeyer. On feedback problems in digraphs. In Proc. WG 1989, volume 411 of Lecture Notes Comput. Sci., pages 218-231. Springer, 1990.
Anke van Zuylen. Linear programming based approximation algorithms for feedback set problems in bipartite tournaments. Theor. Comput. Sci., 412(23):2556-2561, 2011.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Scheduling Distributed Clusters of Parallel Machines: Primal-Dual and LP-based Approximation Algorithms
The Map-Reduce computing framework rose to prominence with datasets of such size that dozens of machines on a single cluster were needed for individual jobs. As datasets approach the exabyte scale, a single job may need distributed processing not only on multiple machines, but on multiple clusters. We consider a scheduling problem to minimize weighted average completion time of n jobs on m distributed clusters of parallel machines. In keeping with the scale of the problems motivating this work, we assume that (1) each job is divided into m "subjobs" and (2) distinct subjobs of a given job may be processed concurrently.
When each cluster is a single machine, this is the NP-Hard concurrent open shop problem. A clear limitation of such a model is that a serial processing assumption sidesteps the issue of how different tasks of a given subjob might be processed in parallel. Our algorithms explicitly model clusters as pools of resources and effectively overcome this issue.
Under a variety of parameter settings, we develop two constant factor approximation algorithms for this problem. The first algorithm uses an LP relaxation tailored to this problem from prior work. This LP-based algorithm provides strong performance guarantees. Our second algorithm exploits a surprisingly simple mapping to the special case of one machine per cluster. This mapping-based algorithm is combinatorial and extremely fast. These are the first constant factor approximations for this problem.
approximation algorithms
distributed computing
machine scheduling
LP relaxations
primal-dual algorithms
68:1-68:17
Regular Paper
Riley
Murray
Riley Murray
Megan
Chao
Megan Chao
Samir
Khuller
Samir Khuller
10.4230/LIPIcs.ESA.2016.68
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http://dx.doi.org/10.1007/978-3-642-14165-2
Zhi-Long Chen and Nicholas G. Hall. Supply chain scheduling: Assembly systems. Working paper., 2000. URL: http://dx.doi.org/10.1007/978-3-8349-8667-2.
http://dx.doi.org/10.1007/978-3-8349-8667-2
Naveen Garg, Amit Kumar, and Vinayaka Pandit. Order Scheduling Models: Hardness and Algorithms. FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science, 4855:96-107, 2007. URL: http://dx.doi.org/10.1007/978-3-540-77050-3_8.
http://dx.doi.org/10.1007/978-3-540-77050-3_8
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Mohammad Hajjat, Shankaranarayanan P N, David Maltz, Sanjay Rao, and Kunwadee Sripanidkulchai. Dealer : Application-aware Request Splitting for Interactive Cloud Applications. CoNEXT 2012, pages 157-168, 2012.
Chien-Chun Hung, Leana Golubchik, and Minlan Yu. Scheduling jobs across geo-distributed datacenters. In Proceedings of the Sixth ACM Symposium on Cloud Computing, pages 111-124. ACM, 2015.
J. Y T Leung, Haibing Li, and Michael Pinedo. Scheduling orders for multiple product types to minimize total weighted completion time. Discrete Applied Mathematics, 155(8):945-970, 2007. URL: http://dx.doi.org/10.1016/j.dam.2006.09.012.
http://dx.doi.org/10.1016/j.dam.2006.09.012
Monaldo Mastrolilli, Maurice Queyranne, Andreas S. Schulz, Ola Svensson, and Nelson A. Uhan. Minimizing the sum of weighted completion times in a concurrent open shop. Operations Research Letters, 38(5):390-395, 2010. URL: http://dx.doi.org/10.1016/j.orl.2010.04.011.
http://dx.doi.org/10.1016/j.orl.2010.04.011
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Riley Murray, Megan Chao, and Samir Khuller. Scheduling distributed clusters of parallel machines [full version], 2016. Unpublished. URL: http://www.cs.umd.edu/users/samir/grant/ESA2016Full.pdf.
http://www.cs.umd.edu/users/samir/grant/ESA2016Full.pdf
Maurice Queyranne. Structure of a simple scheduling polyhedron. Mathematical Programming, 58(1-3):263-285, 1993. URL: http://dx.doi.org/10.1007/BF01581271.
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Andreas S. Schulz. Polytopes and scheduling. PhD Thesis, 1996.
Andreas S Schulz. From linear programming relaxations to approximation algorithms for scheduling problems : A tour d'horizon. Working paper; available upon request., 2012.
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Qiang Zhang, Weiwei Wu, and Minming Li. Resource Scheduling with Supply Constraint and Linear Cost. COCOA 2012 Conference, 2012. http://arxiv.org/abs/9780201398298, URL: http://dx.doi.org/10.1007/3-540-68339-9_34.
http://dx.doi.org/10.1007/3-540-68339-9_34
Creative Commons Attribution 3.0 Unported license
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Finding Large Set Covers Faster via the Representation Method
The worst-case fastest known algorithm for the Set Cover problem on universes with n elements still essentially is the simple O^*(2^n)-time dynamic programming algorithm, and no non-trivial consequences of an O^*(1.01^n)-time algorithm are known. Motivated by this chasm, we study the following natural question: Which instances of Set Cover can we solve faster than the simple dynamic programming algorithm? Specifically, we give a Monte Carlo algorithm that determines the existence of a set cover of size sigma*n in O^*(2^{(1-Omega(sigma^4))n}) time. Our approach is also applicable to Set Cover instances with exponentially many sets: By reducing the task of finding the chromatic number chi(G) of a given n-vertex graph G to Set Cover in the natural way, we show there is an O^*(2^{(1-Omega(sigma^4))n})-time randomized algorithm that given integer s = sigma*n, outputs NO if chi(G) > s and YES with constant probability if \chi(G) <= s - 1.
On a high level, our results are inspired by the "representation method" of Howgrave-Graham and Joux~[EUROCRYPT'10] and obtained by only evaluating a randomly sampled subset of the table entries of a dynamic programming algorithm.
Set Cover
Exact Exponential Algorithms
Fine-Grained Complexity
69:1-69:15
Regular Paper
Jesper
Nederlof
Jesper Nederlof
10.4230/LIPIcs.ESA.2016.69
Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Subset sum in the absence of concentration. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 48-61. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://www.dagstuhl.de/dagpub/978-3-939897-78-1, URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.48.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.48
Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Dense subset sum may be the hardest. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, volume 47 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.13.
http://dx.doi.org/10.4230/LIPIcs.STACS.2016.13
Anja Becker, Jean-Sébastien Coron, and Antoine Joux. Improved generic algorithms for hard knapsacks. In Kenneth G. Paterson, editor, Advances in Cryptology - EUROCRYPT 2011 - 30th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tallinn, Estonia, May 15-19, 2011. Proceedings, volume 6632 of Lecture Notes in Computer Science, pages 364-385. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-20465-4_21.
http://dx.doi.org/10.1007/978-3-642-20465-4_21
Anja Becker, Antoine Joux, Alexander May, and Alexander Meurer. Decoding random binary linear codes in 2 n/20: How 1 + 1 = 0 improves information set decoding. In EUROCRYPT, volume 7237 of Lecture Notes in Computer Science, pages 520-536. Springer, 2012. Talk at http://www.iacr.org/cryptodb/data/paper.php?pubkey=24271. URL: http://dx.doi.org/10.1007/978-3-642-29011-4_31.
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Andreas Björklund. Uniquely coloring graphs over path decompositions. CoRR, abs/1504.03670, 2015. URL: http://arxiv.org/abs/1504.03670.
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Andreas Björklund, Holger Dell, and Thore Husfeldt. The parity of set systems under random restrictions with applications to exponential time problems. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 231-242. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_19.
http://dx.doi.org/10.1007/978-3-662-47672-7_19
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. CoRR, abs/1007.1161, 2010. URL: http://arxiv.org/abs/1007.1161.
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Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Trimmed moebius inversion and graphs of bounded degree. Theory Comput. Syst., 47(3):637-654, 2010. URL: http://dx.doi.org/10.1007/s00224-009-9185-7.
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Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. The traveling salesman problem in bounded degree graphs. ACM Transactions on Algorithms, 8(2):18, 2012. URL: http://dx.doi.org/10.1145/2151171.2151181.
http://dx.doi.org/10.1145/2151171.2151181
Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009. URL: http://dx.doi.org/10.1137/070683933.
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Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. A duality between clause width and clause density for SAT. In 21st Annual IEEE Conference on Computational Complexity (CCC 2006), 16-20 July 2006, Prague, Czech Republic, pages 252-260. IEEE Computer Society, 2006. URL: http://dx.doi.org/10.1109/CCC.2006.6.
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Timothy M. Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1246-1255. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
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Marek Cygan and Marcin Pilipczuk. Faster exponential-time algorithms in graphs of bounded average degree. Inf. Comput., 243:75-85, 2015. URL: http://dx.doi.org/10.1016/j.ic.2014.12.007.
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Vilhelm Dahllöf. Exact Algorithms for Exact Satisfiability Problems. PhD thesis, Linköping University, TCSLAB, The Institute of Technology, 2006.
Evgeny Dantsin, Andreas Goerdt, Edward A Hirsch, Ravi Kannan, Jon Kleinberg, Christos Papadimitriou, Prabhakar Raghavan, and Uwe Schöning. A deterministic (2-2/(k+1))n algorithm for k-SAT based on local search. Theoretical Computer Science, 289(1):69-83, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00174-8.
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Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Homomorphic hashing for sparse coefficient extraction. In Dimitrios M. Thilikos and Gerhard J. Woeginger, editors, Parameterized and Exact Computation - 7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceedings, volume 7535 of Lecture Notes in Computer Science, pages 147-158. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33293-7_15.
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http://dx.doi.org/10.1007/978-3-642-11269-0_21
Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for cnf formulas of bounded modular treewidth. Algorithmica, pages 1-27, 2015. URL: http://dx.doi.org/10.1007/s00453-015-0030-x.
http://dx.doi.org/10.1007/s00453-015-0030-x
Sigve Hortemo Sæther, Jan Arne Telle, and Martin Vatshelle. Solving maxsat and #sat on structured CNF formulas. In Carsten Sinz and Uwe Egly, editors, Theory and Applications of Satisfiability Testing - SAT 2014, Vienna, Austria, July 14-17, 2014. Proceedings, volume 8561 of Lecture Notes in Computer Science, pages 16-31. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-09284-3_3.
http://dx.doi.org/10.1007/978-3-319-09284-3_3
Uwe Schöning. A probabilistic algorithm for k-SAT based on limited local search and restart. Algorithmica, 32(4):615-623, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0094-7.
http://dx.doi.org/10.1007/s00453-001-0094-7
Vijay V. Vazirani. Approximation Algorithms. Springer-Verlag New York, Inc., New York, NY, USA, 2001.
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Graph Isomorphism for Unit Square Graphs
In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.
graph isomorphism
geometric graphs
unit squares
70:1-70:17
Regular Paper
Daniel
Neuen
Daniel Neuen
10.4230/LIPIcs.ESA.2016.70
László Babai. On the automorphism groups of strongly regular graphs I. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 359-368. ACM, 2014. URL: http://dl.acm.org/citation.cfm?id=2554797, URL: http://dx.doi.org/10.1145/2554797.2554830.
http://dx.doi.org/10.1145/2554797.2554830
László Babai. Graph isomorphism in quasipolynomial time. CoRR, abs/1512.03547, 2015. URL: http://arxiv.org/abs/1512.03547.
http://arxiv.org/abs/1512.03547
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http://dx.doi.org/10.1007/978-3-642-40450-4_13
Heinz Breu. Algorithmic Aspects of Constrained Unit Disk Graphs. PhD thesis, University of British Columbia, Vancouver, Canada, 1996.
Heinz Breu and David G. Kirkpatrick. Unit disk graph recognition is np-hard. Comput. Geom., 9(1-2):3-24, 1998. URL: http://dx.doi.org/10.1016/S0925-7721(97)00014-X.
http://dx.doi.org/10.1016/S0925-7721(97)00014-X
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http://dx.doi.org/10.1016/0012-365X(90)90358-O
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Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1563-1575. SIAM, 2012. URL: http://dx.doi.org/10.1137/1.9781611973099.
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Johannes Köbler, Sebastian Kuhnert, Bastian Laubner, and Oleg Verbitsky. Interval graphs: Canonical representations in logspace. SIAM J. Comput., 40(5):1292-1315, 2011. URL: http://dx.doi.org/10.1137/10080395X.
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Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky. Solving the canonical representation and star system problems for proper circular-arc graphs in logspace. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, volume 18 of LIPIcs, pages 387-399. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012. URL: http://www.dagstuhl.de/dagpub/978-3-939897-47-7, URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2012.387.
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Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky. Helly circular-arc graph isomorphism is in logspace. In Krishnendu Chatterjee and Jirí Sgall, editors, Mathematical Foundations of Computer Science 2013 - 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings, volume 8087 of Lecture Notes in Computer Science, pages 631-642. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40313-2_56.
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Bastian Laubner. Capturing polynomial time on interval graphs. In Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, United Kingdom, pages 199-208. IEEE Computer Society, 2010. URL: http://dx.doi.org/10.1109/LICS.2010.42.
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The Alternating Stock Size Problem and the Gasoline Puzzle
Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefixes of the ordering are non-negative, and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. referred to this problem as the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms.
We consider a related problem, which we call the alternating stock size problem, where the number of positive and negative integers in the input set S are equal. The problem is the same as above, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79.
Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.
approximation algorithms
stock size problem
scheduling with non-renewable resources
71:1-71:16
Regular Paper
Alantha
Newman
Alantha Newman
Heiko
Röglin
Heiko Röglin
Johanna
Seif
Johanna Seif
10.4230/LIPIcs.ESA.2016.71
H. M. Abdel-Wahab and T. Kameda. Scheduling to minimize maximum cumulative cost subject to series-parallel precedence constraints. Operations Research, 26(1):141-158, 1978.
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Péter Györgyi and Tamás Kis. Approximation schemes for single machine scheduling with non-renewable resource constraints. Journal of Scheduling, 17(2):135-144, 2014.
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Alantha Newman, Heiko Röglin, and Johanna Seif. The alternating stock size problem and the gasoline puzzle. arXiv:1511.09259, 2015.
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New Parameterized Algorithms for APSP in Directed Graphs
All Pairs Shortest Path (APSP) is a classic problem in graph theory. While for general weighted graphs there is no algorithm that computes APSP in O(n^{3-epsilon}) time (epsilon > 0), by using fast matrix multiplication algorithms, we can compute APSP in O(n^{omega}*log(n)) time (omega < 2.373) for undirected unweighted graphs, and in O(n^{2.5302}) time for directed unweighted graphs. In the current state of matters, there is a substantial gap between the upper bounds of the problem for undirected and directed graphs, and for a long time, it is remained an important open question whether it is possible to close this gap.
In this paper we introduce a new parameter that measures the symmetry of directed graphs (i.e. their closeness to undirected graphs), and obtain a new parameterized APSP algorithm for directed unweighted graphs, that generalizes Seidel's O(n^{omega}*log(n)) time algorithm for undirected unweighted graphs. Given a directed unweighted graph G, unless it is highly asymmetric, our algorithms can compute APSP in o(n^{2.5}) time for G, providing for such graphs a faster APSP algorithm than the state-of-the-art algorithms for the problem.
Graphs
distances
APSP
fast matrix multiplication
72:1-72:13
Regular Paper
Ely
Porat
Ely Porat
Eduard
Shahbazian
Eduard Shahbazian
Roei
Tov
Roei Tov
10.4230/LIPIcs.ESA.2016.72
Donald Aingworth, Chandra Chekuri, Piotr Indyk, and Rajeev Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput., 28(4):1167-1181, 1999. URL: http://dx.doi.org/10.1137/S0097539796303421.
http://dx.doi.org/10.1137/S0097539796303421
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http://dx.doi.org/10.1137/08071990X
Timothy M. Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1246-1255, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
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Liam Roditty and Roei Tov. Approximating the girth. ACM Transactions on Algorithms, 9(2):15, 2013. URL: http://dx.doi.org/10.1145/2438645.2438647.
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Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 515-524, 2013. URL: http://dx.doi.org/10.1145/2488608.2488673.
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http://dx.doi.org/10.1007/PL00009198
Tadao Takaoka. A faster algorithm for the all-pairs shortest path problem and its application. In Computing and Combinatorics, 10th Annual International Conference, COCOON 2004, Jeju Island, Korea, August 17-20, 2004, Proceedings, pages 278-289, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27798-9_31.
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http://dx.doi.org/10.1006/jagm.2000.1080
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http://dx.doi.org/10.1109/FOCS.2010.67
Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM, 49(3):289-317, 2002. URL: http://dx.doi.org/10.1145/567112.567114.
http://dx.doi.org/10.1145/567112.567114
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http://dx.doi.org/10.1007/978-3-540-30551-4_78
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Online Budgeted Maximum Coverage
We study the Online Budgeted Maximum Coverage (OBMC) problem. Subsets of a weighted ground set U arrive one by one, where each set has a cost. The online algorithm has to select a collection of sets, under the constraint that their cost is at most a given budget. Upon arrival of a set the algorithm must decide whether to accept or to reject the arriving set, and it may also drop previously accepted sets (preemption). Rejecting or dropping a set is irrevocable. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection.
We present a deterministic 4/(1-r)-competitive algorithm for OBMC, where r is the maximum ratio between the cost of a set and
the total budget. Building on that algorithm, we then present a randomized O(1)-competitive algorithm for OBMC. On the other hand, we show that the competitive ratio of any deterministic online algorithm is Omega(1/(sqrt{1-r})).
We also give a deterministic O(Delta)-competitive algorithm, where Delta is the maximum weight of a set (given that the minimum element weight is 1), and if the total weight of all elements, w(U), is known in advance, we show that a slight modification of that algorithm is O(min{Delta,sqrt{w(U)}})-competitive. A matching lower bound of Omega(min{Delta,sqrt{w(U)}}) is also given.
Previous to the present work, only the unit cost version of OBMC was studied under the online setting, giving a 4-competitive algorithm [Saha, Getoor, 2009]. Finally, our results, including the lower bounds, apply to Removable Online Knapsack which is the preemptive version of the Online Knapsack problem.
budgeted coverage
maximum coverage
online algorithms
competitive analysis
removable online knapsack
73:1-73:17
Regular Paper
Dror
Rawitz
Dror Rawitz
Adi
Rosén
Adi Rosén
10.4230/LIPIcs.ESA.2016.73
Alexander A. Ageev and Maxim Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8(3):307-328, 2004.
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Giorgio Ausiello, Nicolas Boria, Aristotelis Giannakos, Giorgio Lucarelli, and Vangelis Th. Paschos. Online maximum k-coverage. Discrete Applied Mathematics, 160(13-14):1901-1913, 2012.
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Marek Cygan, Lukasz Jeż, and Jiri Sgall. Online knapsack revisited. Theory of Computing Systems, 2016. To appear.
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Kazuo Iwama and Shiro Taketomi. Removable online knapsack problems. In 29th Annual Intl. Colloquium on Automata, Languages and Programming, volume 2380 of LNCS, pages 293-305, 2002.
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Creative Commons Attribution 3.0 Unported license
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Min-Sum Scheduling Under Precedence Constraints
In many scheduling situations, it is important to consider non-linear functions of job completions times in the objective. This was already recognized by Smith (1956). Recently, the theory community has begun a thorough study of the resulting problems, mostly on single-machine instances for which all permutations of jobs are feasible. However, a typical feature of many scheduling problems is that some jobs can only be processed after others. In this paper, we give the first approximation algorithms for min-sum scheduling with (nonnegative, non-decreasing) non-linear functions and general precedence constraints. In particular, for 1|prec|sum w_j f(C_j), we propose a polynomial-time universal algorithm that performs well for all functions f simultaneously. Its approximation guarantee is 2 for all concave functions, at worst. We also provide a (non-universal) polynomial-time algorithm for the more general case 1|prec|sum f_j(C_j). The performance guarantee is no worse than 2+epsilon for all concave functions. Our results match the best bounds known for the case of linear functions, a widely studied problem, and considerably extend the results for minimizing sum w_jf(C_j) without precedence constraints.
scheduling
approximation algorithms
linear programming relaxations
precedence constraints
74:1-74:13
Regular Paper
Andreas S.
Schulz
Andreas S. Schulz
José
Verschae
José Verschae
10.4230/LIPIcs.ESA.2016.74
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The Power of Migration for Online Slack Scheduling
We investigate the power of migration in online scheduling for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. Once we decide to accept a job, we have to complete it before its deadline d that satisfies d >= (1+epsilon)p + r, where p is the processing time, r the submission time and the slack epsilon > 0 a system parameter. Typically, the hard case arises for small slack epsilon << 1, i.e. for near-tight deadlines. Without migration, a greedy acceptance policy is known to be an optimal deterministic online algorithm with a competitive factor of (1+epsilon)/epsilon (DasGupta and Palis, APPROX 2000). Our first contribution is to show that migrations do not improve the competitive ratio of the greedy acceptance policy, i.e. the competitive ratio remains (1+epsilon)/epsilon for any number of machines.
Our main contribution is a deterministic online algorithm with almost tight competitive ratio on any number of machines. For a single machine, the competitive factor matches the optimal bound of (1+epsilon)/epsilon of the greedy acceptance policy. The competitive ratio improves with an increasing number of machines. It approaches (1+epsilon) ln((1+epsilon)/epsilon) as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small epsilon. Moreover, we show a matching lower bound on the competitive ratio for deterministic algorithms on any number of machines.
Online scheduling
deadlines
preemption with migration
competitive analysis
75:1-75:17
Regular Paper
Chris
Schwiegelshohn
Chris Schwiegelshohn
Uwe
Schwiegelshohn
Uwe Schwiegelshohn
10.4230/LIPIcs.ESA.2016.75
S. Albers and M. Hellwig. On the value of job migration in online makespan minimization. In Proc. of ESA, pages 84-95, 2012.
J. H. Anderson, V. Bud, and U. C. Devi. An EDF-based restricted-migration scheduling algorithm for multiprocessor soft real-time systems. Real-Time Systems, 38(2):85-131, 2008.
S. K. Baruah, G. Koren, D. Mao, B. Mishra, A. Raghunathan, L. E. Rosier, D. Shasha, and F. Wang. On the competitiveness of on-line real-time task scheduling. Real-Time Systems, 4(2):125-144, 1992.
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Sampling-Based Bottleneck Pathfinding with Applications to Fréchet Matching
We describe a general probabilistic framework to address a variety of Fréchet-distance optimization problems. Specifically, we are interested in finding minimal bottleneck-paths in d-dimensional Euclidean space between given start and goal points, namely paths that minimize the maximal value over a continuous cost map. We present an efficient and simple sampling-based framework for this problem, which is inspired by, and draws ideas from, techniques for robot motion planning. We extend the framework to handle not only standard bottleneck pathfinding, but also the more demanding case, where the path needs to be monotone in all dimensions. Finally, we provide experimental results of the framework on several types of problems.
Computational geometry
Fréchet distances
sampling-based algorithms
random geometric graphs
bottleneck pathfinding
76:1-76:16
Regular Paper
Kiril
Solovey
Kiril Solovey
Dan
Halperin
Dan Halperin
10.4230/LIPIcs.ESA.2016.76
Aviv Adler, Mark de Berg, Dan Halperin, and Kiril Solovey. Efficient multi-robot motion planning for unlabeled discs in simple polygons. IEEE Trans. Automation Science and Engineering, 12(4):1309-1317, 2015.
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Paul Balister, Amites Sarkar, and Béla Bollobás. Percolation, connectivity, coverage and colouring of random geometric graphs. In Béla Bollobás, Robert Kozma, and Dezso Miklós, editors, Handbook of Large-Scale Random Networks, pages 117-142. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.
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Kevin Buchin, Maike Buchin, Maximilian Konzack, Wolfgang Mulzer, and André Schulz. Fine-grained analysis of problems on curves. In EuroCG, Lugano, Switzerland, 2016.
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Kevin Buchin, Maike Buchin, Rolf van Leusden, Wouter Meulemans, and Wolfgang Mulzer. Computing the Fréchet distance with a retractable leash. In European Symposium of Algorithms, pages 241-252, 2013.
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Shiri Chechik, Haim Kaplan, Mikkel Thorup, Or Zamir, and Uri Zwick. Bottleneck paths and trees and deterministic graphical games. In Symposium on Theoretical Aspects of Computer Science, pages 27:1-27:13, 2016.
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Sariel Har-Peled and Benjamin Raichel. The Fréchet distance revisited and extended. ACM Transactions on Algorithms, 10(1):3, 2014.
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On the Geodesic Centers of Polygonal Domains
In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain P of n vertices. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of P that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the size of the shortest path map equivalence decomposition of P, which is known to be O(n^{10}). One key observation is a pi-range property on shortest path lengths when points are moving. With these observations, we propose an algorithm that computes all geodesic centers in O(n^{11}*log(n)) time. Previously, an algorithm of O(n^{12+epsilon}) time was known for this problem, for any epsilon > 0.
geodesic centers
shortest paths
polygonal domains
77:1-77:17
Regular Paper
Haitao
Wang
Haitao Wang
10.4230/LIPIcs.ESA.2016.77
H.-K. Ahn, L. Barba, P. Bose, J.-L. De Carufel, M. Korman, and E. Oh. A linear-time algorithm for the geodesic center of a simple polygon. In Proc. of the 31st Annual Symposium on Computational Geometry (SoCG), pages 209-223, 2015.
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S.W. Bae, M. Korman, Y. Okamoto, and H. Wang. Computing the L₁ geodesic diameter and center of a simple polygon in linear time. Computational Geometry: Theory and Applications, 48:495-505, 2015.
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The Complexity of the k-means Method
The k-means method is a widely used technique for clustering points in Euclidean space. While it is extremely fast in practice, its worst-case running time is exponential in the number of data points. We prove that the k-means method can implicitly solve PSPACE-complete problems, providing a complexity-theoretic explanation for its worst-case running time. Our result parallels recent work on the complexity of the simplex method for linear programming.
k-means
PSPACE-complete
78:1-78:14
Regular Paper
Tim
Roughgarden
Tim Roughgarden
Joshua R.
Wang
Joshua R. Wang
10.4230/LIPIcs.ESA.2016.78
Ilan Adler, Christos Papadimitriou, and Aviad Rubinstein. On simplex pivoting rules and complexity theory. In Integer Programming and Combinatorial Optimization, pages 13-24. Springer, 2014.
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