15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016), SWAT 2016, June 22-24, 2016, Reykjavik, Iceland
SWAT 2016
June 22-24, 2016
Reykjavik, Iceland
Scandinavian Symposium and Workshops on Algorithm Theory
SWAT
https://dblp.org/db/conf/swat
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Rasmus
Pagh
Rasmus Pagh
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
53
2016
978-3-95977-011-8
https://www.dagstuhl.de/dagpub/978-3-95977-011-8
Front Matter, Table of Contents, Preface, Program Committee, Subreviewers
Front Matter, Table of Contents, Preface, Program Committee, Subreviewers
Front Matter
Table of Contents
Preface
Program Committee
Subreviewers
0:i-0:xiv
Front Matter
Rasmus
Pagh
Rasmus Pagh
10.4230/LIPIcs.SWAT.2016.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding
We consider a lower- and upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds).
We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well.
Facility Location
Approximation Algorithm
LP Rounding
1:1-1:14
Regular Paper
Zachary
Friggstad
Zachary Friggstad
Mohsen
Rezapour
Mohsen Rezapour
Mohammad R.
Salavatipour
Mohammad R. Salavatipour
10.4230/LIPIcs.SWAT.2016.1
Zoë Abrams, Adam Meyerson, Kamesh Munagala, and Serge Plotkin. The integrality gap of capacitated facility location. Technical Report CMU-CS-02-199, Carnegie Mellon University, 2002.
Ankit Aggarwal, Anand Louis, Manisha Bansal, Naveen Garg, Neelima Gupta, Shubham Gupta, and Surabhi Jain. A 3-approximation algorithm for the facility location problem with uniform capacities. Mathematical Programming, 141, 2013.
Sara Ahmadian and Chaitanya Swamy. Improved approximation guarantees for lower-bounded facility location. In proceedings of WAOA 2012, pages 257-271, 2012.
Hyung-Chan An, Monika Singh, and Ola Svensson. LP-based algorithms for capacitated facility location. In proceedings of FOCS 2014, pages 256-265, 2014.
Manisha Bansal, Naveen Garg, and Neelima Gupta. A 5-approximation for capacitated facility location. In proceedings of ESA 2012, pages 133-144, 2012.
Fabián A Chudak and David P Williamson. Improved approximation algorithms for capacitated facility location problems. Mathematical programming, 102, 2005.
Friedrich Eisenbrand, Fabrizio Grandoni, Thomas Rothvoß, and Guido Schäfer. Connected facility location via random facility sampling and core detouring. Journal of Computer and System Sciences, 76(8):709-726, 2010.
Fabrizio Grandoni and Thomas Rothvoß. Approximation algorithms for single and multi-commodity connected facility location. In proceedings of IPCO 2011, pages 248-260, 2011.
Sudipto Guha, Adam Meyerson, and Kamesh Munagala. Hierarchical placement and network design problems. In proceedings of FOCS 2000, pages 603-612, 2000.
Anupam Gupta, Jon Kleinberg, Amit Kumar, Rajeev Rastogi, and Bulent Yener. Provisioning a virtual private network: a network design problem for multicommodity flow. In proceedings of STOC 2001, pages 389-398, 2001.
Hyunwoo Jung, Mohammad Khairul Hasan, and Kyung-Yong Chwa. A 6.55 factor primal-dual approximation algorithm for the connected facility location problem. Journal of combinatorial optimization, 18(3):258-271, 2009.
DR Karget and Maria Minkoff. Building steiner trees with incomplete global knowledge. In proceedings of FOCS 2000, pages 613-623, 2000.
Madhukar R Korupolu, C Greg Plaxton, and Rajmohan Rajaraman. Analysis of a local search heuristic for facility location problems. Journal of algorithms, 2000.
Retsef Levi, David B Shmoys, and Chaitanya Swamy. LP-based approximation algorithms for capacitated facility location. Mathematical programming, 131, 2012.
Shi Li. On uniform capacitated k-median beyond the natural lp relaxation. In proceedings of SODA 2015, pages 696-707, 2015.
Shi Li. Approximating capacitated k-median with (1+ε)k open facilities. In proceedings of SODA 2016, 2016.
Martin Pal, Eva Tardos, and Tom Wexler. Facility location with nonuniform hard capacities. In proceedings of FOCS 2001, pages 329-338, 2001.
David B Shmoys, Éva Tardos, and Karen Aardal. Approximation algorithms for facility location problems. In proceedings of STOC 1997, pages 265-274, 1997.
Zoya Svitkina. Lower-bounded facility location. ACM Transactions on Algorithms (TALG), 6(4):69, 2010.
Chaitanya Swamy and Amit Kumar. Primal-dual algorithms for connected facility location problems. Algorithmica, 40(4):245-269, 2004.
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Approximation Algorithms for Node-Weighted Prize-Collecting Steiner Tree Problems on Planar Graphs
We study the prize-collecting version of the node-weighted Steiner tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a 2.88-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev [6] for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain [17].
approximation algorithms
Node-Weighted Steiner Tree
primal-dual algorithm
LMP
planar graphs
2:1-2:14
Regular Paper
Jaroslaw
Byrka
Jaroslaw Byrka
Mateusz
Lewandowski
Mateusz Lewandowski
Carsten
Moldenhauer
Carsten Moldenhauer
10.4230/LIPIcs.SWAT.2016.2
Ajit Agrawal, Philip N. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized steiner problem on networks. SIAM J. Comput., 24(3):440-456, 1995.
Aaron Archer, MohammadHossein Bateni, MohammadTaghi Hajiaghayi, and Howard J. Karloff. Improved approximation algorithms for prize-collecting steiner tree and TSP. SIAM J. Comput., 40(2):309-332, 2011.
MohammadHossein Bateni, Chandra Chekuri, Alina Ene, Mohammad Taghi Hajiaghayi, Nitish Korula, and Dániel Marx. Prize-collecting steiner problems on planar graphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1028-1049, 2011.
MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Dániel Marx. Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth. J. ACM, 58(5):21, 2011.
MohammadHossein Bateni, MohammadTaghi Hajiaghayi, and Vahid Liaghat. Improved approximation algorithms for (budgeted) node-weighted steiner problems. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 81-92, 2013.
Piotr Berman and Grigory Yaroslavtsev. Primal-dual approximation algorithms for node-weighted network design in planar graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, pages 50-60, 2012.
Jarosław Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. An improved lp-based approximation for steiner tree. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 583-592, 2010.
Jarosław Byrka, Mateusz Lewandowski, and Carsten Moldenhauer. Approximation algorithms for node-weighted prize-collecting steiner tree problems on planar graphs. CoRR, abs/1601.02481, 2016.
Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Philip N. Klein. Node-weighted steiner tree and group steiner tree in planar graphs. ACM Trans. Algorithms, 10(3):13:1-13:20, 2014.
Bistra N. Dilkina and Carla P. Gomes. Solving connected subgraph problems in wildlife conservation. In Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 7th International Conference, CPAIOR 2010, Bologna, Italy, June 14-18, 2010. Proceedings, pages 102-116, 2010.
Karoline Faust, Pierre Dupont, Jérôme Callut, and Jacques van Helden. Pathway discovery in metabolic networks by subgraph extraction. Bioinformatics, 26(9):1211-1218, 2010.
M. R. Garey and David S. Johnson. The rectilinear steiner tree problem in NP complete. SIAM Journal of Applied Mathematics, 32:826-834, 1977.
Joseph Geunes, Retsef Levi, H. Edwin Romeijn, and David B. Shmoys. Approximation algorithms for supply chain planning and logistics problems with market choice. Math. Program., 130(1):85-106, 2011.
Michel X. Goemans. Combining approximation algorithms for the prize-collecting TSP. CoRR, abs/0910.0553, 2009.
Michel X. Goemans and David P. Williamson. A general approximation technique for constrained forest problems. SIAM J. Comput., 24(2):296-317, 1995.
Sudipto Guha, Anna Moss, Joseph Naor, and Baruch Schieber. Efficient recovery from power outage (extended abstract). In Proc. of the 31st Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA, pages 574-582, 1999.
Mohammad Taghi Hajiaghayi and Kamal Jain. The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema. In Proc. of the 7th Annual ACM-SIAM Symp. on Discrete Algorithms, SODA'06, pages 631-640, 2006.
Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001.
Jochen Könemann, Sina Sadeghian Sadeghabad, and Laura Sanità. An LMP o(log n)-approximation algorithm for node weighted prize collecting steiner tree. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 568-577, 2013.
Carsten Moldenhauer. Primal-dual approximation algorithms for node-weighted steiner forest on planar graphs. Inf. Comput., 222:293-306, 2013.
Carsten Moldenhauer. Node-weighted network design and maximum sub-determinants. PhD thesis, EPFL, 2014.
David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.
David Paul Williamson. On the design of approximation algorithms for a class of graph problems. PhD thesis, MIT, Cambridge, MA, September 1993.
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A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs
We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is O(log k) in quasi-bipartite graphs with k terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor. A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs.
Approximation algorithm
Primal-Dual algorithm
Directed Steiner tree
3:1-3:11
Regular Paper
Zachary
Friggstad
Zachary Friggstad
Jochen
Könemann
Jochen Könemann
Mohammad
Shadravan
Mohammad Shadravan
10.4230/LIPIcs.SWAT.2016.3
J. Byrka, F. Grandoni, T. Rothvoss, , and L. Sanita. Steiner tree approximation via iterative randomized rounding. Journal of the ACM, 60(1), 2013.
G. Calinescu and G. Zelikovsky. The polymatroid steiner problems. J. Combinatorial Optimization, 9(3):281-294, 2005.
M. Charikar, C. Chekuri, T. Cheung, Z. Dai, A. Goel, S. Guha, , and M. Li. Approximation algorithms for directed steiner problems. J. Algorithms, 33(1):73-91, 1999.
I. Dinur and D. Steurer. Analytic approach to parallel repetition. In In proceedings of STOC, 2014.
J. Edmonds. Optimum branchings. J. Res. Natl. Bur. Stand., 71:233-240, 1967.
U. Feige. A threshold of ln n for approximating set-cover. Journal of the ACM, 45(4):634-652, 1998.
Z. Friggstad, A. Louis, Y. K. Ko, J. Könemann, M. Shadravan, and M. Tulsiani. Linear programming hierarchies suffice for directed steiner tree. In In proceedings of IPCO, 2014.
M. X. Goemans, N. Olver, T. Rothvoss, and R. Zenklusen. Matroids and integrality gaps for hypergraphic steiner tree relaxations. In In proceedings of STOC, 2012.
M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24(2):296-317, 1995.
S. Guha, A. Moss, J. Naor, and B. Scheiber. Efficient recover from power outage. In In proceedings of STOC, 1999.
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T. Hibi and T. Fujito. Multi-rooted greedy approximation of directed steiner trees with applications. In In proceedings of WG, 2012.
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L. Zosin and S. Khuller. On directed steiner trees. In In proceedings of SODA, 2002.
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A Linear Kernel for Finding Square Roots of Almost Planar Graphs
A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph is a graph that can be made planar by the removal of at most k vertices. We prove that the generalization of Square Root, in which we are given two subsets of edges prescribed to be in or out of a square root, respectively, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.
planar graphs
square roots
linear kernel
4:1-4:14
Regular Paper
Petr A.
Golovach
Petr A. Golovach
Dieter
Kratsch
Dieter Kratsch
Daniël
Paulusma
Daniël Paulusma
Anthony
Stewart
Anthony Stewart
10.4230/LIPIcs.SWAT.2016.4
Anna Adamaszek and Michal Adamaszek. Uniqueness of graph square roots of girth six. Electronic Journal of Combinatorics, 18, 2011.
Manfred Cochefert, Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. Sparse square roots. In Andreas Brandstädt, Klaus Jansen, and Rüdiger Reischuk, editors, Graph-Theoretic Concepts in Computer Science - 39th International Workshop, WG 2013, Lübeck, Germany, June 19-21, 2013, Revised Papers, volume 8165 of Lecture Notes in Computer Science, pages 177-188. Springer, 2013.
Manfred Cochefert, Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. Parameterized algorithms for finding square roots. Algorithmica, 74:602-629, 2016.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate Texts in Mathematics. Springer, 2012.
Babak Farzad and Majid Karimi. Square-root finding problem in graphs, a complete dichotomy theorem. CoRR, abs/1210.7684, 2012.
Babak Farzad, Lap Chi Lau, Van Bang Le, and Nguyen Ngoc Tuy. Complexity of finding graph roots with girth conditions. Algorithmica, 62:38-53, 2012.
Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. Squares of low clique number. In 14th Cologne Twente Workshop 2016 (CTW 2016), Gargnano, Italy, June 6-8, 2016, Electronic Notes in Discrete Mathematics, to appear, 2016.
Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984.
Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: distance from triviality. In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3162 of Lecture Notes in Computer Science, pages 162-173. Springer, 2004.
Frank Harary, Richard M. Karp, and William T. Tutte. A criterion for planarity of the square of a graph. Journal of Combinatorial Theory, 2:395-405, 1967.
Lap Chi Lau. Bipartite roots of graphs. ACM Transactions on Algorithms, 2:178-208, 2006.
Lap Chi Lau and Derek G. Corneil. Recognizing powers of proper interval, split, and chordal graphs. SIAM Journal on Discrete Mathematics, 18:83-102, 2004.
Van Bang Le, Andrea Oversberg, and Oliver Schaudt. Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs. Theoretical Computer Science, 602:39-49, 2015.
Van Bang Le, Andrea Oversberg, and Oliver Schaudt. A unified approach for recognizing squares of split graphs. Manuscript, 2015.
Van Bang Le and Nguyen Ngoc Tuy. The square of a block graph. Discrete Mathematics, 310:734-741, 2010.
Van Bang Le and Nguyen Ngoc Tuy. A good characterization of squares of strongly chordal split graphs. Information Processing Letters, 111:120-123, 2011.
Yaw-Ling Lin and Steven Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8:99-118, 1995.
Martin Milanic, Andrea Oversberg, and Oliver Schaudt. A characterization of line graphs that are squares of graphs. Discrete Applied Mathematics, 173:83-91, 2014.
Martin Milanic and Oliver Schaudt. Computing square roots of trivially perfect and threshold graphs. Discrete Applied Mathematics, 161:1538-1545, 2013.
Rajeev Motwani and Madhu Sudan. Computing roots of graphs is hard. Discrete Applied Mathematics, 54:81-88, 1994.
A. Mukhopadhyay. The square root of a graph. Journal of Combinatorial Theory, 2:290-295, 1967.
Nestor V. Nestoridis and Dimitrios M. Thilikos. Square roots of minor closed graph classes. Discrete Applied Mathematics, 168:34-39, 2014.
Ian C. Ross and Frank Harary. The square of a tree. Bell System Technical Journal, 39:641-647, 1960.
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Linear-Time Recognition of Map Graphs with Outerplanar Witness
Map graphs generalize planar graphs and were introduced by Chen, Grigni and Papadimitriou [STOC 1998, J.ACM 2002]. They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup [FOCS 1998] published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Omega(n^{120}) for n-vertex graphs, and a full description of its details remains unpublished.
We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup's approach, it computes the witness graph W in the affirmative case.
Algorithms and data structures
map graphs
recognition
planar graphs
5:1-5:14
Regular Paper
Matthias
Mnich
Matthias Mnich
Ignaz
Rutter
Ignaz Rutter
Jens M.
Schmidt
Jens M. Schmidt
10.4230/LIPIcs.SWAT.2016.5
F. J. Brandenburg. On 4-map graphs and 1-planar graphs and their recognition problem. Technical report, ArXiv, 2015. URL: http://arxiv.org/abs/1509.03447.
http://arxiv.org/abs/1509.03447
Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou. Map graphs. J. ACM, 49(2):127-138, 2002.
Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou. Recognizing hole-free 4-map graphs in cubic time. Algorithmica, 45(2):227-262, 2006.
Z.-Z. Chen, X. He, and M.-Y. Kao. Nonplanar topological inference and political-map graphs. In Proc. SODA 1999, pages 195-204, 1999.
E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and D. M. Thilikos. Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs. ACM Trans. Algorithms, 1(1):33-47, 2005.
G. Di Battista and R. Tamassia. On-Line Maintenance of Triconnected Components with SPQR-Trees. Algorithmica, 15(4):302-318, 1996.
G. Di Battista and R. Tamassia. On-Line Planarity Testing. SIAM J. Comput., 25(5):956-997, 1996.
F. V. Fomin, D. Lokshtanov, N. Misra, and S. Saurabh. Planar ℱ-deletion: Approximation, kernelization and optimal FPT algorithms. Proc. FOCS 2012, pages 470-479, 2012.
F. V. Fomin, D. Lokshtanov, and S. Saurabh. Bidimensionality and geometric graphs. In Proc. SODA 2012, pages 1563-1575, 2012.
A. Grigoriev and H. L. Bodlaender. Algorithms for graphs embeddable with few crossings per edge. Algorithmica, 49(1):1-11, 2007. URL: http://dx.doi.org/10.1007/s00453-007-0010-x.
http://dx.doi.org/10.1007/s00453-007-0010-x
C. Gutwenger and P. Mutzel. A linear time implementation of SPQR-trees. In Proc. GD 2000, volume 1984 of Lecture Notes Comput. Sci., pages 77-90, 2001.
J. Hopcroft and R. Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, 1974.
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V. P. Korzhik and B. Mohar. Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory, 72(1):30-71, 2013.
Y. L. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM J. Discrete Math., 8(1):99-118, 1995.
R. M. McConnell, K. Mehlhorn, S. Näher, and P. Schweitzer. Certifying algorithms. Comput. Sci. Review, 5(2):119-161, 2011.
M. M. Sysło. Characterizations of outerplanar graphs. Discrete Math., 26(1):47-53, 1979.
M. Thorup. Map graphs in polynomial time. In Proc. FOCS 1998, pages 396-405, 1998.
K. Wagner. Über eine Eigenschaft der ebenen Komplexe. Math. Ann., 114(1):570-590, 1937.
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The p-Center Problem in Tree Networks Revisited
We present two improved algorithms for weighted discrete p-center problem for tree networks with n vertices. One of our proposed algorithms runs in O(n*log(n) + p*log^2(n) * log(n/p)) time. For all values of p, our algorithm thus runs as fast as or faster than the most efficient O(n*log^2(n)) time algorithm obtained by applying Cole's [1987] speed-up technique to the algorithm due to Megiddo and Tamir [1983], which has remained unchallenged for nearly 30 years.
Our other algorithm, which is more practical, runs in O(n*log(n) + p^2*log^2(n/p)) time, and when p=O(sqrt(n)) it is faster than Megiddo and Tamir's O(n*log^2(n) * log(log(n))) time algorithm [1983].
Facility location
p-center
parametric search
tree network
sorting network
6:1-6:15
Regular Paper
Aritra
Banik
Aritra Banik
Binay
Bhattacharya
Binay Bhattacharya
Sandip
Das
Sandip Das
Tsunehiko
Kameda
Tsunehiko Kameda
Zhao
Song
Zhao Song
10.4230/LIPIcs.SWAT.2016.6
M. Ajtai, J. Komlós, and E. Szemerédi. An O(n log n) sorting network. In Proc. 15th ACM Symp. on Theory of Comput. (STOC), pages 1-9, 1983.
I. Averbakh and O. Berman. Minimax regret p-center location on a network with demand uncertainty. Location Science, 5:247-254, 1997.
Boaz Ben-Moshe, Binay Bhattacharya, and Qiaosheng Shi. An optimal algorithm for the continuous/discrete weighted 2-center problem in trees. In Proc. LATIN 2006, volume LNCS 3887, pages 166-177, 2006.
R. Benkoczi. Cardinality constrained facility location problems in trees. PhD thesis, School of Computing Science, Simon Fraser University, Canada, 2004.
R. Benkoczi, B. Bhattacharya, M. Chrobak, L. Larmore, and W. Rytter. Faster algorithms for k-median problems in trees. Mathematical Foundations of Computer Science, Springer-Verlag, LNCS 2747:218-227, 2003.
Binay Bhattacharya, Tsunehiko Kameda, and Zhao Song. Minmax regret 1-center on a path/cycle/tree. In Proc. 6th Int'l Conf. on Advanced Engineering Computing and Applications in Sciences (ADVCOMP), pages 108-113, 2012.
Binay Bhattacharya and Qiaosheng Shi. Improved algorithms to network p-center location problems. Computational Geometry, 47:307-315, 2014.
Timothy M. Chan. Klee’s measure problem made easy. In Proc. Symp. on Foundation of Computer Science (FOCS), pages 410-419, 2013.
Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1:133-162, 1986.
R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34:200-208, 1987.
G.N. Frederickson. Optimal algorithms for partitioning trees and locating p centers in trees. Technical Report CSD-TR-1029, Purdue University, 1990.
G.N. Frederickson. Parametric search and locating supply centers in trees. In Proc. Workshop on Algorithms and Data Structures (WADS), Springer-Verlag, volume LNCS 519, pages 299-319, 1991.
Michael T. Goodrich. Zig-zag sort: A simple deterministic data-oblivious sorting algorithm running in O(nlog n) time. arXiv:1403,2777v1 [cs.DS] 11 Mar2014, 2014.
S.L. Hakimi. Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research, 12:450-459, 1964.
Trevor S. Hale and Christopher R. Moberg. Location science research: A review. Annals of Operations Research, 123:21-35, 2003.
M. Jeger and O. Kariv. Algorithms for finding p-centers on a weighted tree (for relatively small p). Networks, 15:381-389, 1985.
O. Kariv and S.L. Hakimi. An algorithmic approach to network location problems, part 1: The p-centers. SIAM J. Appl. Math., 37:513-538, 1979.
N. Megiddo. Combinatorial optimization with rational objective functions. Math. Oper. Res., 4:414-424, 1979.
N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. ACM, 30:852-865, 1983.
N. Megiddo and A. Tamir. New results on the complexity of p-center problems. SIAM J. Comput., 12:751-758, 1983.
N. Megiddo, A. Tamir, E. Zemel, and R. Chandrasekaran. An O(nlog² n) algorithm for the kth longest path in a tree with applications to location problems. SIAM J. Comput., 10:328-337, 1981.
M.S. Paterson. Improved sorting networks with O(log n) depth. Algorithmica, 5:75-92, 1990.
Joel Seiferas. Sorting networks of logarithmic depth, further simplified. Algorithmica, 53:374-384, 2009.
Q. Shi. Efficient algorithms for network center/covering location optimization problems. PhD thesis, School of Computing Science, Simon Fraser University, Canada, 2008.
A. Tamir. Improved complexity bounds for center location problems on networks by using dynamic structures. SIAM J. Discrete Mathematics, 1:377-396, 1988.
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A Simple Mergeable Dictionary
A mergeable dictionary is a data structure storing a dynamic subset S of a totally ordered set U and supporting predecessor searches in S. Apart from insertions and deletions to S, we can both merge two arbitrarily interleaved dictionaries and split a given dictionary around some pivot x. We present an implementation of a mergeable dictionary matching the optimal amortized logarithmic bounds of Iacono and Ökzan [Iacono/Ökzan,ICALP'10]. However, our solution is significantly simpler. The proposed data structure can also be generalized to the case when the universe U is dynamic or infinite, thus addressing one issue of [Iacono/Ökzan,ICALP'10].
dictionary
mergeable
data structure
merge
split
7:1-7:13
Regular Paper
Adam
Karczmarz
Adam Karczmarz
10.4230/LIPIcs.SWAT.2016.7
Lars Arge and Jeffrey Scott Vitter. Optimal external memory interval management. SIAM J. Comput., 32(6):1488-1508, 2003. URL: http://dx.doi.org/10.1137/S009753970240481X.
http://dx.doi.org/10.1137/S009753970240481X
Amitabha Bagchi, Adam L. Buchsbaum, and Michael T. Goodrich. Biased skip lists. Algorithmica, 42(1):31-48, 2005. URL: http://dx.doi.org/10.1007/s00453-004-1138-6.
http://dx.doi.org/10.1007/s00453-004-1138-6
Michael A. Bender, Richard Cole, Erik D. Demaine, Martin Farach-Colton, and Jack Zito. Two simplified algorithms for maintaining order in a list. In Algorithms - ESA 2002, 10th Annual European Symposium, Rome, Italy, September 17-21, 2002, Proceedings, pages 152-164, 2002. URL: http://dx.doi.org/10.1007/3-540-45749-6_17.
http://dx.doi.org/10.1007/3-540-45749-6_17
Mark R. Brown and Robert Endre Tarjan. A fast merging algorithm. J. ACM, 26(2):211-226, 1979. URL: http://dx.doi.org/10.1145/322123.322127.
http://dx.doi.org/10.1145/322123.322127
Paul F. Dietz and Daniel Dominic Sleator. Two algorithms for maintaining order in a list. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 365-372, 1987. URL: http://dx.doi.org/10.1145/28395.28434.
http://dx.doi.org/10.1145/28395.28434
Martin Farach and Mikkel Thorup. String matching in lempel-ziv compressed strings. Algorithmica, 20(4):388-404, 1998. URL: http://dx.doi.org/10.1007/PL00009202.
http://dx.doi.org/10.1007/PL00009202
Paweł Gawrychowski. Pattern matching in lempel-ziv compressed strings: Fast, simple, and deterministic. In Algorithms - ESA 2011 - 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings, pages 421-432, 2011. URL: http://dx.doi.org/10.1007/978-3-642-23719-5_36.
http://dx.doi.org/10.1007/978-3-642-23719-5_36
Loukas Georgiadis, Haim Kaplan, Nira Shafrir, Robert Endre Tarjan, and Renato Fonseca F. Werneck. Data structures for mergeable trees. ACM Transactions on Algorithms, 7(2):14, 2011. URL: http://dx.doi.org/10.1145/1921659.1921660.
http://dx.doi.org/10.1145/1921659.1921660
Leonidas J. Guibas and Robert Sedgewick. A dichromatic framework for balanced trees. In 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16-18 October 1978, pages 8-21, 1978. URL: http://dx.doi.org/10.1109/SFCS.1978.3.
http://dx.doi.org/10.1109/SFCS.1978.3
Scott Huddleston and Kurt Mehlhorn. A new data structure for representing sorted lists. Acta Inf., 17:157-184, 1982. URL: http://dx.doi.org/10.1007/BF00288968.
http://dx.doi.org/10.1007/BF00288968
John Iacono and Özgür Özkan. Mergeable dictionaries. In Proceedings of the 37th International Colloquium Conference on Automata, Languages and Programming, ICALP'10, pages 164-175, Berlin, Heidelberg, 2010. Springer-Verlag. URL: http://dx.doi.org/10.1007/978-3-642-14165-2_15.
http://dx.doi.org/10.1007/978-3-642-14165-2_15
Tsvi Kopelowitz. On-line indexing for general alphabets via predecessor queries on subsets of an ordered list. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 283-292, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.79.
http://dx.doi.org/10.1109/FOCS.2012.79
Katherine Jane Lai. Complexity of union-split-find problems. Master’s thesis, Massachusetts Institute of Technology, 2008. URL: http://erikdemaine.org/theses/klai.pdf.
http://erikdemaine.org/theses/klai.pdf
William Pugh. Skip lists: A probabilistic alternative to balanced trees. Commun. ACM, 33(6):668-676, 1990. URL: http://dx.doi.org/10.1145/78973.78977.
http://dx.doi.org/10.1145/78973.78977
Mihai Pătraşcu and Erik D. Demaine. Lower bounds for dynamic connectivity. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 546-553, 2004. URL: http://dx.doi.org/10.1145/1007352.1007435.
http://dx.doi.org/10.1145/1007352.1007435
Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26(3):362-391, June 1983. URL: http://dx.doi.org/10.1016/0022-0000(83)90006-5.
http://dx.doi.org/10.1016/0022-0000(83)90006-5
Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary trees. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 235-245, 1983. URL: http://dx.doi.org/10.1145/800061.808752.
http://dx.doi.org/10.1145/800061.808752
Dan E. Willard. Log-logarithmic worst-case range queries are possible in space theta(n). Inf. Process. Lett., 17(2):81-84, 1983. URL: http://dx.doi.org/10.1016/0020-0190(83)90075-3.
http://dx.doi.org/10.1016/0020-0190(83)90075-3
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Cuckoo Filter: Simplification and Analysis
The cuckoo filter data structure of Fan, Andersen, Kaminsky, and Mitzenmacher (CoNEXT 2014) performs the same approximate set operations as a Bloom filter in less memory, with better locality of reference, and adds the ability to delete elements as well as to insert them. However, until now it has lacked theoretical guarantees on its performance. We describe a simplified version of the cuckoo filter using fewer hash function calls per query. With this simplification, we provide the first theoretical performance guarantees on cuckoo filters, showing that they succeed with high probability whenever their fingerprint length is large enough.
approximate set
Bloom filter
cuckoo filter
cuckoo hashing
8:1-8:12
Regular Paper
David
Eppstein
David Eppstein
10.4230/LIPIcs.SWAT.2016.8
Martin Aumüller, Martin Dietzfelbinger, and Philipp Woelfel. Explicit and efficient hash families suffice for cuckoo hashing with a stash. Algorithmica, 70(3):428-456, 2014. URL: http://dx.doi.org/10.1007/s00453-013-9840-x.
http://dx.doi.org/10.1007/s00453-013-9840-x
Burton H. Bloom. Space/time trade-offs in hash coding with allowable errors. Commun. ACM, 13(7):422-426, 1970. URL: http://dx.doi.org/10.1145/362686.362692.
http://dx.doi.org/10.1145/362686.362692
Graham Cormode and S. Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. J. Algorithms, 55(1):58-75, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2003.12.001.
http://dx.doi.org/10.1016/j.jalgor.2003.12.001
Martin Dietzfelbinger and Ulf Schellbach. On risks of using cuckoo hashing with simple universal hash classes. In Proc. 20th ACM-SIAM Symp. Discrete Algorithms (SODA'09), pages 795-804, 2009.
Martin Dietzfelbinger and Christoph Weidling. Balanced allocation and dictionaries with tightly packed constant size bins. Theoret. Comput. Sci., 380(1-2):47-68, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.02.054.
http://dx.doi.org/10.1016/j.tcs.2007.02.054
David Eppstein and Michael T. Goodrich. Straggler identification in round-trip data streams via Newton’s identities and invertible Bloom filters. IEEE Trans. Knowledge and Data Engineering, 23(2):297-306, 2011. http://arxiv.org/abs/0704.3313, URL: http://dx.doi.org/10.1109/TKDE.2010.132.
http://dx.doi.org/10.1109/TKDE.2010.132
David Eppstein, Michael T. Goodrich, Frank Uyeda, and George Varghese. What’s the difference? Efficient set reconciliation without prior context. In Proc. ACM SIGCOMM 2011, pages 218-229, 2011. URL: http://dx.doi.org/10.1145/2018436.2018462.
http://dx.doi.org/10.1145/2018436.2018462
Bin Fan, Dave G. Andersen, Michael Kaminsky, and Michael D. Mitzenmacher. Cuckoo filter: Practically better than Bloom. In Proc. 10th ACM Int. Conf. Emerging Networking Experiments and Technologies (CoNEXT'14), pages 75-88, 2014. URL: http://dx.doi.org/10.1145/2674005.2674994.
http://dx.doi.org/10.1145/2674005.2674994
Li Fan, Pei Cao, Jussara Almeida, and Andrei Broder. Summary cache: A scalable wide-area web cache sharing protocol. IEEE/ACM Trans. Networking, 8(3):281-293, 2000. URL: http://dx.doi.org/10.1109/90.851975.
http://dx.doi.org/10.1109/90.851975
M. Grissa, A.A. Yavuz, and B. Hamdaoui. Cuckoo filter-based location-privacy preservation in database-driven cognitive radio networks. In Proc. World Symp. Computer Networks and Information Security (WSCNIS 2015), pages 1-7. IEEE, 2015. URL: http://dx.doi.org/10.1109/WSCNIS.2015.7368280.
http://dx.doi.org/10.1109/WSCNIS.2015.7368280
Vikas Gupta and Frank Breitinger. How cuckoo filter can improve existing approximate matching techniques. In Joshua I. James and Frank Breitinger, editors, Proc. 7th Int. Conf. Digital Forensics and Cyber Crime (ICDF2C 2015), volume 157 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, pages 39-52. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-25512-5_4.
http://dx.doi.org/10.1007/978-3-319-25512-5_4
Adam Kirsch, Michael D. Mitzenmacher, and Udi Wieder. More robust hashing: cuckoo hashing with a stash. SIAM J. Comput., 39(4):1543-1561, 2010. URL: http://dx.doi.org/10.1137/080728743.
http://dx.doi.org/10.1137/080728743
Anna Pagh, Rasmus Pagh, and S. Srinivasa Rao. An optimal Bloom filter replacement. In Proc. 16th ACM-SIAM Symposium on Discrete Algorithms (SODA'05), pages 823-829. ACM, New York, 2005.
Rasmus Pagh and Flemming Friche Rodler. Cuckoo hashing. J. Algorithms, 51(2):122-144, 2004. URL: http://dx.doi.org/10.1016/j.jalgor.2003.12.002.
http://dx.doi.org/10.1016/j.jalgor.2003.12.002
Mihai Pătraşcu and Mikkel Thorup. The power of simple tabulation hashing. J. ACM, 59(3):A14, 2012. URL: http://dx.doi.org/10.1145/2220357.2220361.
http://dx.doi.org/10.1145/2220357.2220361
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Randomized Algorithms for Finding a Majority Element
Given n colored balls, we want to detect if more than n/2 of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to minimize the total number of such operations. A well-known exercise is to show how to find such a ball with only 2n comparisons while using only a logarithmic number of bits for bookkeeping. The resulting algorithm is called the Boyer-Moore majority vote algorithm. It is known that any deterministic method needs 3n/2-2 comparisons in the worst case, and this is tight. However, it is not clear what is the required number of comparisons if we allow randomization. We construct a randomized algorithm which always correctly finds a ball of the majority color (or detects that there is none) using, with high probability, only 7n/6+o(n) comparisons. We also prove that the expected number of comparisons used by any such randomized method is at least 1.019n.
majority
randomized algorithms
lower bounds
9:1-9:14
Regular Paper
Pawel
Gawrychowski
Pawel Gawrychowski
Jukka
Suomela
Jukka Suomela
Przemyslaw
Uznanski
Przemyslaw Uznanski
10.4230/LIPIcs.SWAT.2016.9
Martin Aigner, Gianluca De Marco, and Manuela Montangero. The plurality problem with three colors and more. Theor. Comput. Sci., 337(1-3):319-330, 2005.
Laurent Alonso, Edward M. Reingold, and René Schott. Determining the majority. Inf. Process. Lett., 47(5):253-255, 1993.
Laurent Alonso, Edward M. Reingold, and René Schott. The average-case complexity of determining the majority. SIAM J. Comput., 26(1):1-14, 1997.
Robert S. Boyer and J. Strother Moore. MJRTY: A fast majority vote algorithm. In Automated Reasoning: Essays in Honor of Woody Bledsoe, pages 105-118, 1991.
Demetres Christofides. On randomized algorithms for the majority problem. Discrete Applied Mathematics, 157(7):1481-1485, 2009.
Fan R. K. Chung, Ronald L. Graham, Jia Mao, and Andrew Chi-Chih Yao. Oblivious and adaptive strategies for the majority and plurality problems. Algorithmica, 48(2):147-157, 2007.
Dorit Dor and Uri Zwick. Selecting the median. SIAM J. Comput., 28(5):1722-1758, 1999.
Dorit Dor and Uri Zwick. Median selection requires (2+ε)n comparisons. SIAM J. Discrete Math., 14(3):312-325, 2001.
David Eppstein and Daniel S. Hirschberg. From discrepancy to majority. In LATIN, volume 9644 of Lecture Notes in Computer Science, pages 390-402. Springer, 2016.
P. Erdős. On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc., 51(12):898-902, 12 1945.
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Dániel Gerbner, Gyula O. H. Katona, Dömötör Pálvölgyi, and Balázs Patkós. Majority and plurality problems. Discrete Applied Mathematics, 161(6):813-818, 2013.
Daniel Král, Jirí Sgall, and Tomás Tichý. Randomized strategies for the plurality problem. Discrete Applied Mathematics, 156(17):3305-3311, 2008.
Gianluca De Marco and Evangelos Kranakis. Searching for majority with k-tuple queries. Discrete Math., Alg. and Appl., 7(2), 2015.
Gianluca De Marco and Andrzej Pelc. Randomized algorithms for determining the majority on graphs. Combinatorics, Probability & Computing, 15(6):823-834, 2006.
Mike Paterson. Progress in selection. In Algorithm Theory-SWAT'96, pages 368-379. Springer, 1996.
Michael E. Saks and Michael Werman. On computing majority by comparisons. Combinatorica, 11(4):383-387, 1991.
Máté Vizer, Dániel Gerbner, Balázs Keszegh, Dömötör Pálvölgyi, Balázs Patkós, and Gábor Wiener. Finding a majority ball with majority answers. Electronic Notes in Discrete Mathematics, 49:345-351, 2015.
Gábor Wiener. Search for a majority element. Journal of Statistical Planning and Inference, 100(2):313-318, 2002.
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A Framework for Dynamic Parameterized Dictionary Matching
Two equal-length strings S and S' are a parameterized-match (p-match) iff there exists a one-to-one function that renames the characters in S to those in S'. Let P be a collection of d patterns of total length n characters that are chosen from an alphabet Sigma of cardinality sigma. The task is to index P such that we can support the following operations.
* search(T): given a text T, report all occurrences <j,P_i> such that there exists a pattern P_i in P that is a p-match with the substring T[j,j+|P_i|-1].
* ins(P_i)/del(P_i): modify the index when a pattern P_i is inserted/deleted.
We present a linear-space index that occupies O(n*log n) bits and supports (i) search(T) in worst-case O(|T|*log^2 n + occ) time, where occ is the number of occurrences reported, and (ii) ins(P_i) and del(P_i) in amortized O(|P_i|*polylog(n)) time.
Then, we present a succinct index that occupies (1+o(1))n*log sigma + O(d*log n) bits and supports (i) search(T) in worst-case O(|T|*log^2 n + occ) time, and (ii) ins(P_i) and del(P_i) in amortized O(|P_i|*polylog(n)) time. We also present results related to the semi-dynamic variant of the problem, where deletion is not allowed.
Parameterized Dictionary Indexing
Generalized Suffix Tree
Succinct Data Structures
Sparsification
10:1-10:14
Regular Paper
Arnab
Ganguly
Arnab Ganguly
Wing-Kai
Hon
Wing-Kai Hon
Rahul
Shah
Rahul Shah
10.4230/LIPIcs.SWAT.2016.10
Alfred V. Aho and Margaret J. Corasick. Efficient string matching: An aid to bibliographic search. Commun. ACM, 18(6):333-340, 1975. URL: http://dx.doi.org/10.1145/360825.360855.
http://dx.doi.org/10.1145/360825.360855
Stephen Alstrup, Thore Husfeldt, and Theis Rauhe. Marked ancestor problems. In 39th Annual Symposium on Foundations of Computer Science, FOCS'98, November 8-11, 1998, Palo Alto, California, USA, pages 534-544, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743504.
http://dx.doi.org/10.1109/SFCS.1998.743504
Amihood Amir, Martin Farach, Zvi Galil, Raffaele Giancarlo, and Kunsoo Park. Dynamic dictionary matching. J. Comput. Syst. Sci., 49(2):208-222, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80047-9.
http://dx.doi.org/10.1016/S0022-0000(05)80047-9
Amihood Amir, Martin Farach, Ramana M. Idury, Johannes A. La Poutré, and Alejandro A. Schäffer. Improved dynamic dictionary matching. Inf. Comput., 119(2):258-282, 1995. URL: http://dx.doi.org/10.1006/inco.1995.1090.
http://dx.doi.org/10.1006/inco.1995.1090
Brenda S. Baker. A theory of parameterized pattern matching: algorithms and applications. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 71-80, 1993. URL: http://dx.doi.org/10.1145/167088.167115.
http://dx.doi.org/10.1145/167088.167115
Djamal Belazzougui. Succinct dictionary matching with no slowdown. In Combinatorial Pattern Matching, 21st Annual Symposium, CPM 2010, New York, NY, USA, June 21-23, 2010. Proceedings, pages 88-100, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13509-5_9.
http://dx.doi.org/10.1007/978-3-642-13509-5_9
Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In LATIN 2000: Theoretical Informatics, 4th Latin American Symposium, Punta del Este, Uruguay, April 10-14, 2000, Proceedings, pages 88-94, 2000. URL: http://dx.doi.org/10.1007/10719839_9.
http://dx.doi.org/10.1007/10719839_9
Sudip Biswas, Arnab Ganguly, Rahul Shah, and Sharma V. Thankachan. Forbidden extension queries. In 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, pages 320-335, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.320.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.320
Ho-Leung Chan, Wing-Kai Hon, Tak Wah Lam, and Kunihiko Sadakane. Dynamic dictionary matching and compressed suffix trees. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 13-22, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070436.
http://dl.acm.org/citation.cfm?id=1070432.1070436
Paul F. Dietz and Daniel Dominic Sleator. Two algorithms for maintaining order in a list. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 365-372, 1987. URL: http://dx.doi.org/10.1145/28395.28434.
http://dx.doi.org/10.1145/28395.28434
Martin Dietzfelbinger, Anna R. Karlin, Kurt Mehlhorn, Friedhelm Meyer auf der Heide, Hans Rohnert, and Robert Endre Tarjan. Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738-761, 1994. URL: http://dx.doi.org/10.1137/S0097539791194094.
http://dx.doi.org/10.1137/S0097539791194094
Guy Feigenblat, Ely Porat, and Ariel Shiftan. An improved query time for succinct dynamic dictionary matching. In Combinatorial Pattern Matching - 25th Annual Symposium, CPM 2014, Moscow, Russia, June 16-18, 2014. Proceedings, pages 120-129, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07566-2_13.
http://dx.doi.org/10.1007/978-3-319-07566-2_13
Paolo Ferragina and Giovanni Manzini. Opportunistic data structures with applications. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12-14 November 2000, Redondo Beach, California, USA, pages 390-398, 2000. URL: http://dx.doi.org/10.1109/SFCS.2000.892127.
http://dx.doi.org/10.1109/SFCS.2000.892127
Arnab Ganguly, Rahul Shah, and Sharma V. Thankachan. Succinct non-overlapping indexing. In Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Ischia Island, Italy, June 29 - July 1, 2015, Proceedings, pages 185-195, 2015. URL: http://dx.doi.org/10.1007/978-3-319-19929-0_16.
http://dx.doi.org/10.1007/978-3-319-19929-0_16
Roberto Grossi, Ankur Gupta, and Jeffrey Scott Vitter. High-order entropy-compressed text indexes. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 841-850, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644250.
http://dl.acm.org/citation.cfm?id=644108.644250
Roberto Grossi and Jeffrey Scott Vitter. Compressed suffix arrays and suffix trees with applications to text indexing and string matching (extended abstract). In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 397-406, 2000. URL: http://dx.doi.org/10.1145/335305.335351.
http://dx.doi.org/10.1145/335305.335351
Wing-Kai Hon, Tak Wah Lam, Rahul Shah, Siu-Lung Tam, and Jeffrey Scott Vitter. Compressed index for dictionary matching. In 2008 Data Compression Conference (DCC 2008), 25-27 March 2008, Snowbird, UT, USA, pages 23-32, 2008. URL: http://dx.doi.org/10.1109/DCC.2008.62.
http://dx.doi.org/10.1109/DCC.2008.62
Wing-Kai Hon, Tak Wah Lam, Rahul Shah, Siu-Lung Tam, and Jeffrey Scott Vitter. Succinct index for dynamic dictionary matching. In Algorithms and Computation, 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings, pages 1034-1043, 2009. URL: http://dx.doi.org/10.1007/978-3-642-10631-6_104.
http://dx.doi.org/10.1007/978-3-642-10631-6_104
Ramana M. Idury and Alejandro A. Schäffer. Multiple matching of parameterized patterns. In Combinatorial Pattern Matching, 5th Annual Symposium, CPM 94, Asilomar, California, USA, June 5-8, 1994, Proceedings, pages 226-239, 1994. URL: http://dx.doi.org/10.1007/3-540-58094-8_20.
http://dx.doi.org/10.1007/3-540-58094-8_20
Tsvi Kopelowitz and Moshe Lewenstein. Dynamic weighted ancestors. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 565-574, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283444.
http://dl.acm.org/citation.cfm?id=1283383.1283444
S. Rao Kosaraju. Faster algorithms for the construction of parameterized suffix trees (preliminary version). In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23-25 October 1995, pages 631-637, 1995. URL: http://dx.doi.org/10.1109/SFCS.1995.492664.
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Moshe Lewenstein. Parameterized pattern matching. In Encyclopedia of Algorithms, 2015. URL: http://dx.doi.org/10.1007/978-3-642-27848-8_282-2.
http://dx.doi.org/10.1007/978-3-642-27848-8_282-2
Veli Mäkinen and Gonzalo Navarro. Dynamic entropy-compressed sequences and full-text indexes. ACM Transactions on Algorithms, 4(3), 2008. URL: http://dx.doi.org/10.1145/1367064.1367072.
http://dx.doi.org/10.1145/1367064.1367072
Edward M. McCreight. A space-economical suffix tree construction algorithm. J. ACM, 23(2):262-272, 1976. URL: http://dx.doi.org/10.1145/321941.321946.
http://dx.doi.org/10.1145/321941.321946
J. Ian Munro, Gonzalo Navarro, Jesper Sindahl Nielsen, Rahul Shah, and Sharma V. Thankachan. Top- k term-proximity in succinct space. In Algorithms and Computation - 25th International Symposium, ISAAC 2014, Jeonju, Korea, December 15-17, 2014, Proceedings, pages 169-180, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0_14.
http://dx.doi.org/10.1007/978-3-319-13075-0_14
Gonzalo Navarro and Veli Mäkinen. Compressed full-text indexes. ACM Comput. Surv., 39(1), 2007. URL: http://dx.doi.org/10.1145/1216370.1216372.
http://dx.doi.org/10.1145/1216370.1216372
Gonzalo Navarro and Kunihiko Sadakane. Fully functional static and dynamic succinct trees. ACM Trans. Algorithms, 10(3):16:1-16:39, 2014. URL: http://dx.doi.org/10.1145/2601073.
http://dx.doi.org/10.1145/2601073
Mark H. Overmars. The Design of Dynamic Data Structures, volume 156 of Lecture Notes in Computer Science. Springer, 1983. URL: http://dx.doi.org/10.1007/BFb0014927.
http://dx.doi.org/10.1007/BFb0014927
Kunihiko Sadakane. Compressed text databases with efficient query algorithms based on the compressed suffix array. In Algorithms and Computation, 11th International Conference, ISAAC 2000, Taipei, Taiwan, December 18-20, 2000, Proceedings, pages 410-421, 2000. URL: http://dx.doi.org/10.1007/3-540-40996-3_35.
http://dx.doi.org/10.1007/3-540-40996-3_35
Dekel Tsur. Top-k document retrieval in optimal space. Inf. Process. Lett., 113(12):440-443, 2013. URL: http://dx.doi.org/10.1016/j.ipl.2013.03.012.
http://dx.doi.org/10.1016/j.ipl.2013.03.012
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Efficient Summing over Sliding Windows
This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of Omega(1/epsilon + log(W)) memory bits for Wepsilon-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/epsilon + log(W)) bits, indicating that the algorithm is optimal and that the bound is tight. Next, the more general problem of maintaining a sum of the last W integers, each in the range of {0, 1, ..., R}, is addressed. The paper shows that approximating the sum within an additive error of RW epsilon can also be done using Theta(1/epsilon + log(W)) bits for epsilon = Omega(1/W). For epsilon = o(1/W), we present a succinct algorithm which uses B(1 + o(1)) bits, where B = Theta(W*log(1/(W*epsilon))) is the derived lower bound. We show that all lower bounds generalize to randomized algorithms as well. All algorithms process new elements and answer queries in O(1) worst-case time.
Streaming
Statistics
Lower Bounds
11:1-11:14
Regular Paper
Ran
Ben Basat
Ran Ben Basat
Gil
Einziger
Gil Einziger
Roy
Friedman
Roy Friedman
Yaron
Kassner
Yaron Kassner
10.4230/LIPIcs.SWAT.2016.11
Michael H Albert, Alexander Golynski, Angèle M Hamel, Alejandro López-Ortiz, S Srinivasa Rao, and Mohammad Ali Safari. Longest increasing subsequences in sliding windows. Theoretical Computer Science, 321(2):405-414, 2004.
Arvind Arasu and Gurmeet Singh Manku. Approximate counts and quantiles over sliding windows. In Proc. of the 23rd ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, PODS 2004. Association for Computing Machinery, Inc., June 2004.
Brian Babcock, Mayur Datar, Rajeev Motwani, and Liadan O'Callaghan. Maintaining variance and k-medians over data stream windows. In Frank Neven, Catriel Beeri, and Tova Milo, editors, Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 9-12, 2003, San Diego, CA, USA, pages 234-243. ACM, 2003.
Ran Ben Basat, Gil Einziger, Roy Friedman, and Yaron Kassner. Efficient summing over sliding windows. CoRR, abs/1604.02450, 2016. URL: http://arxiv.org/abs/1604.02450.
http://arxiv.org/abs/1604.02450
Ran Ben Basat, Gil Einziger, Roy Friedman, and Yaron Kassner. Heavy hitters in streams and sliding windows. In INFOCOM, 2016 Proceedings IEEE, pages 307-315, April 2016.
Vladimir Braverman, Ran Gelles, and Rafail Ostrovsky. How to catch l2-heavy-hitters on sliding windows. Theoretical Computer Science, 554:82-94, 2014.
Vladimir Braverman and Rafail Ostrovsky. Smooth histograms for sliding windows. In Foundations of Computer Science, 2007. FOCS'07. 48th Annual IEEE Symposium on, pages 283-293. IEEE, 2007.
Edith Cohen and Martin J. Strauss. Maintaining time-decaying stream aggregates. J. Algorithms, 59(1):19-36, 2006.
Graham Cormode and Ke Yi. Tracking distributed aggregates over time-based sliding windows. In Scientific and Statistical Database Management, pages 416-430. Springer, 2012.
Michael Crouch and Daniel S. Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, pages 96-104, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96
Michael S Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Algorithms-ESA 2013, pages 337-348. Springer, 2013.
Mayur Datar, Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Maintaining stream statistics over sliding windows. SIAM J. Comput., 31(6):1794-1813, 2002.
Phillip B. Gibbons and Srikanta Tirthapura. Distributed streams algorithms for sliding windows. In SPAA, pages 63-72, 2002.
Regant Y.S. Hung and H. F. Ting. Finding heavy hitters over the sliding window of a weighted data stream. In E. Laber, C. Bornstein, L. Nogueira, and L. Faria, editors, LATIN 2008: Theoretical Informatics, volume 4957 of LNCS, pages 699-710. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-78773-0_60.
http://dx.doi.org/10.1007/978-3-540-78773-0_60
Lap-Kei Lee and H. F. Ting. Maintaining significant stream statistics over sliding windows. In Proceedings of the Seventeenth Annual Symposium on Discrete Algorithms, SODA, pages 724-732. ACM Press, 2006.
Lap-Kei Lee and HF Ting. A simpler and more efficient deterministic scheme for finding frequent items over sliding windows. In Proc. of the SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 290-297. ACM, 2006.
Yang Liu, Wenji Chen, and Yong Guan. Near-optimal approximate membership query over time-decaying windows. In INFOCOM, Proceedings IEEE, pages 1447-1455, April 2013.
Kyriakos Mouratidis, Spiridon Bakiras, and Dimitris Papadias. Continuous monitoring of top-k queries over sliding windows. In Proc. of the International Conference on Management of Data, SIGMOD, pages 635-646, New York, NY, USA, 2006. ACM.
Moni Naor and Eylon Yogev. Sliding bloom filters. In Leizhen Cai, Siu-Wing Cheng, and Tak-Wah Lam, editors, Algorithms and Computation, volume 8283 of Lecture Notes in Computer Science, pages 513-523. Springer Berlin Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-45030-3_48.
http://dx.doi.org/10.1007/978-3-642-45030-3_48
Krešimir Pripužić, Ivana Podnar Žarko, and Karl Aberer. Time- and space-efficient sliding window top-k query processing. ACM Trans. Database Syst., 40(1):1:1-1:44, March 2015.
Hong Shen and Yu Zhang. Improved approximate detection of duplicates for data streams over sliding windows. Journal of Computer Science and Technology, 23(6):973-987, 2008.
Zhitao Shen, M.A. Cheema, Xuemin Lin, Wenjie Zhang, and Haixun Wang. Efficiently monitoring top-k pairs over sliding windows. In Data Engineering (ICDE), 2012 IEEE 28th International Conference on, pages 798-809, April 2012.
Andrew Chi-Chin Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symp. on Foundations of Computer Science, pages 222-227. IEEE, 1977.
Wenjie Zhang, Ying Zhang, Muhammad Aamir Cheema, and Xuemin Lin. Counting distinct objects over sliding windows. In Proceedings of the Twenty-First Australasian Conference on Database Technologies - Volume 104, ADC'10, pages 75-84, Darlinghurst, Australia, Australia, 2010. Australian Computer Society, Inc.
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Lower Bounds for Approximation Schemes for Closest String
In the Closest String problem one is given a family S of equal-length strings over some fixed alphabet, and the task is to find a string y that minimizes the maximum Hamming distance between y and a string from S. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [Li et al.; J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that FPT=W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time f(eps) n^o(1/eps), for any computable function f, would contradict the Exponential Time Hypothesis.
closest string
PTAS
efficient PTAS
12:1-12:10
Regular Paper
Marek
Cygan
Marek Cygan
Daniel
Lokshtanov
Daniel Lokshtanov
Marcin
Pilipczuk
Marcin Pilipczuk
Michal
Pilipczuk
Michal Pilipczuk
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.SWAT.2016.12
Alexandr Andoni, Piotr Indyk, and Mihai Pătraşcu. On the optimality of the dimensionality reduction method. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 449-458. IEEE Computer Society, 2006.
Cristina Bazgan. Schémas d'approximation et complexité paramétrée. PhD thesis, Université Paris Sud, 1995. In French.
Christina Boucher, Christine Lo, and Daniel Lokshtanov. Consensus Patterns (probably) has no EPTAS. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 239-250. Springer, 2015. Full version available at URL: http://www.ii.uib.no/~daniello/papers/ConsensusPatterns.pdf.
http://www.ii.uib.no/~daniello/papers/ConsensusPatterns.pdf
Marco Cesati and Luca Trevisan. On the efficiency of polynomial time approximation schemes. Inf. Process. Lett., 64(4):165-171, 1997.
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał 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
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2006.
Jens Gramm, Rolf Niedermeier, and Peter Rossmanith. Fixed-parameter algorithms for Closest String and related problems. Algorithmica, 37(1):25-42, 2003.
Ming Li, Bin Ma, and Lusheng Wang. Finding similar regions in many sequences. J. Comput. Syst. Sci., 65(1):73-96, 2002.
Ming Li, Bin Ma, and Lusheng Wang. On the Closest String and Substring problems. J. ACM, 49(2):157-171, 2002.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 760-776. SIAM, 2011.
Bin Ma and Xiaoming Sun. More efficient algorithms for Closest String and Substring problems. SIAM J. Comput., 39(4):1432-1443, 2009.
Dániel Marx. Closest substring problems with small distances. SIAM J. Comput., 38(4):1382-1410, 2008.
Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008.
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Coloring Graphs Having Few Colorings Over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.
Graph vertex coloring
path decomposition
Alon-Tarsi theorem
13:1-13:9
Regular Paper
Andreas
Björklund
Andreas Björklund
10.4230/LIPIcs.SWAT.2016.13
A. Abboud and V. Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proceedings of the IEEE FOCS, pages 434-443, 2014.
N. Alon and M. Tarsi. Colorings and orientations of graphs. Combinatorica, 12:125-134, 1992.
A. Backurs and P. Indyk. Edit distance cannot be computed in strongly subquadratic time (unless seth is false). In Proceedings of the ACM STOC, pages 51-58, 2015.
R. Barbanchon. On unique graph 3-colorability and parsimonious reductions in the plane. Theoretical Computer Science, 319:455-482, 2004.
A. Björklund, H. Dell, and T. Husfeldt. The parity of set systems under random restrictions with applications to exponential time problems. In Proceedings of ICALP, pages 231-242, 2015.
R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philosophical Society, Math. Phys. Sci, 37:194-197, 1941.
C. Calabro, R. Impagliazzo, V. Kabanets, and R. Paturi. The complexity of unique k-sat: An isolation lemma for k-cnfs. Journal of Computer and System Sciences, 74:386-393, 2008.
M. Cygan, S. Kratsch, and J. Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the ACM STOC, pages 301-310, 2013.
D. Hefetz. On two generalizations of the alon-tarsi polynomial method. Journal of Combinatorial Theory Series B, 101:403-414, 2011.
C. Hillar and T. Windfeldt. An algebraic characterization of uniquely vertex colorable graphs. Journal of Combinatorial Theory Series B, 98:400-414, 2008.
I. Holyer. The np-completeness of edge-coloring. SIAM J. Comput., 10:718-720, 1981.
R. Impagliazzo and R. Paturi. The complexity of k-sat. In Proceedings of the IEEE Computational Complexity Conference, pages 237-240, 1999.
T. B. Jensen and B. Toft. Graph Coloring Problems. Wiley-Interscience, 1st edition, 1994.
R. M. Karp. Reducibility among combinatorial problems. Complexity of Computer Computations, pages 85-103, 1972.
D. Lokshtanov, D. Marx, and S. Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. In Proceedings of ACM-SIAM SODA, pages 777-789, 2011.
B. Reed. A strengthening of brook’s theorem. Journal of Combinatorial Theory Series B, 76:136-149, 1999.
N. Robertson and P. Seymour. Graph minors. i. excluding a forest. Journal of Combinatorial Theory Series B, 35:39-61, 1983.
S. Xu. The size of uniquely colorable graphs. Journal of Combinatorial Theory Series B, 50:319-320, 1990.
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Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs
We consider the recognition problem for two graph classes that generalize split and unipolar graphs, respectively.
First, we consider the recognizability of graphs that admit a monopolar partition: a partition of the vertex set into sets A,B such that G[A] is a disjoint union of cliques and G[B] an independent set. If in such a partition G[A] is a single clique, then G is a split graph. We show that in
O(2^k * k^3 * (|V(G)| + |E(G)|)) time we can decide whether G admits a monopolar partition
(A,B) where G[A] has at most k cliques. This generalizes the linear-time algorithm for recognizing split graphs corresponding to the case when k=1.
Second, we consider the recognizability of graphs that admit a 2-subcoloring: a partition of the vertex set into sets A,B such that each of G[A] and G[B] is a disjoint union of cliques. If in such a partition G[A] is a single clique, then G is a unipolar graph. We show that in
O(k^(2k+2) * (|V(G)|^2+|V(G)| * |E(G)|)) time we can decide whether G admits a
2-subcoloring (A,B) where G[A] has at most k cliques. This generalizes the polynomial-time algorithm for recognizing unipolar graphs corresponding to the case when k=1.
We also show that in O(4^k) time we can decide whether G admits a 2-subcoloring (A,B) where G[A] and G[B] have at most k cliques in total.
To obtain the first two results above, we formalize a technique, which we dub inductive recognition, that can
be viewed as an adaptation of iterative compression to recognition problems. We believe that the formalization
of this technique will prove useful in general for designing parameterized algorithms for recognition problems.
Finally, we show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the
above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the
recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k
cliques.
vertex-partition problems
monopolar graphs
subcolorings
split graphs
unipolar graphs
fixed-parameter algorithms
14:1-14:14
Regular Paper
Iyad
Kanj
Iyad Kanj
Christian
Komusiewicz
Christian Komusiewicz
Manuel
Sorge
Manuel Sorge
Erik
Jan van Leeuwen
Erik Jan van Leeuwen
10.4230/LIPIcs.SWAT.2016.14
Demetrios Achlioptas. The complexity of G-free colourability. Discrete Mathematics, 165–166(0):21-30, 1997.
Hajo Broersma, Fedor V. Fomin, Jaroslav Nešetřil, and Gerhard J. Woeginger. More about subcolorings. Computing, 69(3):187-203, 2002.
Sharon Bruckner, Falk Hüffner, and Christian Komusiewicz. A graph modification approach for finding core-periphery structures in protein interaction networks. Algorithms for Molecular Biology, 10:16, 2015.
Ross Churchley and Jing Huang. List monopolar partitions of claw-free graphs. Discrete Mathematics, 312(17):2545-2549, 2012.
Ross Churchley and Jing Huang. Solving partition problems with colour-bipartitions. Graphs and Combinatorics, 30(2):353-364, 2014.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Reinhard Diestel. Graph Theory, 4th Edition. Springer, 2012.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin, Heidelberg, 2013.
Tınaz Ekim, Pavol Hell, Juraj Stacho, and Dominique de Werra. Polarity of chordal graphs. Discrete Applied Mathematics, 156(13):2469-2479, 2008.
E. M. Eschen and X. Wang. Algorithms for unipolar and generalized split graphs. Discrete Applied Mathematics, 162:195-201, 2014.
Alastair Farrugia. Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard. The Electronic Journal of Combinatorics, 11(1):R46, 2004.
Jirí Fiala, Klaus Jansen, Van Bang Le, and Eike Seidel. Graph subcolorings: Complexity and algorithms. SIAM Journal on Discrete Mathematics, 16(4):635-650, 2003.
Peter L. Hammer and Bruno Simeone. The splittance of a graph. Combinatorica, 1(3):275-284, 1981.
Sudeshna Kolay and Fahad Panolan. Parameterized Algorithms for Deletion to (r,𝓁)-Graphs. In Proceedings of the 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, volume 45 of LIPIcs, pages 420-433. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015.
Van Bang Le and Ragnar Nevries. Complexity and algorithms for recognizing polar and monopolar graphs. Theoretical Computer Science, 528:1-11, 2014.
Colin McDiarmid and Nikola Yolov. Recognition of unipolar and generalised split graphs. Algorithms, 8(1):46-59, 2015.
C. H. Papadimitriou. Computational Complexity. Addison Wesley, 1994.
Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Opererations Research Letters, 32(4):299-301, 2004.
Juraj Stacho. On 2-subcolourings of chordal graphs. In Proceedings of the 8th Latin American Theoretical Informatics Symposium, volume 4957 of LNCS, pages 544-554. Springer, 2008.
Regina I. Tyshkevich and Arkady A. Chernyak. Algorithms for the canonical decomposition of a graph and recognizing polarity. Izvestia Akad. Nauk BSSR, ser. Fiz. Mat. Nauk, 6:16-23, 1985. In Russian.
Regina I. Tyshkevich and Arkady A. Chernyak. Decomposition of graphs. Cybernetics and Systems Analysis, 21(2):231-242, 1985.
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On Routing Disjoint Paths in Bounded Treewidth Graphs
We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph G and a collection of k source-destination pairs M = (s_1, t_1), ..., (s_k, t_k). The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset M' of the pairs is a collection P of paths such that, for each pair (s_i, t_i) in M', there is a path in P connecting s_i to t_i. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph G has capacities cap(e) on the edges and a routing P is feasible if each edge e is in at most cap(e) of the paths of P. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP.
In this paper we obtain an O(r^3) approximation for MaxEDP on graphs of treewidth at most r and a matching approximation for MaxNDP on graphs of pathwidth at most r. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an O(r * 3^r) approximation for MaxEDP.
Algorithms and data structures
disjoint paths
treewidth
15:1-15:15
Regular Paper
Alina
Ene
Alina Ene
Matthias
Mnich
Matthias Mnich
Marcin
Pilipczuk
Marcin Pilipczuk
Andrej
Risteski
Andrej Risteski
10.4230/LIPIcs.SWAT.2016.15
Alexandr Andoni, Anupam Gupta, and Robert Krauthgamer. Towards (1 + ε)-approximate flow sparsifiers. In Proc. SODA 2014, pages 279-293, 2014.
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.
Parinya Chalermsook, Julia Chuzhoy, Alina Ene, and Shi Li. Approximation algorithms and hardness of integral concurrent flow. In Proc. STOC 2012, pages 689-708, 2012.
C. Chekuri, S. Khanna, and F.B. Shepherd. An O(√n) approximation and integrality gap for disjoint paths and unsplittable flow. Theory Comput., 2(7):137-146, 2006.
Chandra Chekuri and Julia Chuzhoy. Large-treewidth graph decompositions and applications. In Proc. STOC 2013, pages 291-300, 2013.
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. Multicommodity flow, well-linked terminals, and routing problems. In Proc. STOC 2005, pages 183-192, 2005.
Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. Edge-disjoint paths in planar graphs with constant congestion. SIAM J. Comput., 39(1):281-301, 2009.
Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. A note on multiflows and treewidth. Algorithmica, 54(3):400-412, 2009.
Chandra Chekuri, Guyslain Naves, and F. Bruce Shepherd. Maximum edge-disjoint paths in k-sums of graphs. In Proc. ICALP 2013, volume 7965 of LNCS, pages 328-339, 2013.
Chandra Chekuri, Guyslain Naves, and F. Bruce Shepherd. Maximum edge-disjoint paths in k-sums of graphs. Technical report, ArXiv, 2013. URL: http://arxiv.org/abs/1303.4897.
http://arxiv.org/abs/1303.4897
Eden Chlamtac, Robert Krauthgamer, and Prasad Raghavendra. Approximating sparsest cut in graphs of bounded treewidth. In Proc. APPROX-RANDOM 2010, volume 6302 of LNCS, pages 124-137, 2010.
Julia Chuzhoy. On vertex sparsifiers with Steiner nodes. In Proc. STOC 2012, pages 673-688, 2012.
Julia Chuzhoy and Shi Li. A polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2. In Proc. FOCS 2012, pages 233-242, 2012.
Reinhard Diestel. Graph theory, volume 173. Springer, third edition, 2005.
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.
Anupam Gupta, Kunal Talwar, and David Witmer. Sparsest cut on bounded treewidth graphs: algorithms and hardness results. In Proc. STOC 2013, pages 281-290, 2013.
R.M. Karp. On the computational complexity of combinatorial problems. Networks, 5:45-68, 1975.
S.G. Kolliopoulos and C. Stein. Approximating disjoint-path problems using packing integer programs. Math. Prog., 99(1):63-87, 2004.
Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012.
Neil Robertson and Paul D Seymour. Graph minors. V. Excluding a planar graph. J. Combinatorial Theory, Ser. B, 41(1):92-114, 1986.
Neil Robertson and Paul D Seymour. Graph minors. XIII. The disjoint paths problem. J. Combinatorial Theory, Ser. B, 63(1):65-110, 1995.
Neil Robertson and Paul D Seymour. Graph minors. XVI. Excluding a non-planar graph. J. Combinatorial Theory, Ser. B, 89(1):43-76, 2003.
Löc Séguin-Charbonneau and F Bruce Shepherd. Maximum edge-disjoint paths in planar graphs with congestion 2. In Proc. FOCS 2011, pages 200-209, 2011.
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Colouring Diamond-free Graphs
The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.
colouring
clique-width
diamond-free
graph class
hereditary graph class
16:1-16:14
Regular Paper
Konrad K.
Dabrowski
Konrad K. Dabrowski
François
Dross
François Dross
Daniël
Paulusma
Daniël Paulusma
10.4230/LIPIcs.SWAT.2016.16
Claudio Arbib and Raffaele Mosca. On (P₅,diamond)-free graphs. Discrete Mathematics, 250(1-3):1-22, 2002.
Rodica Boliac and Vadim V. Lozin. On the clique-width of graphs in hereditary classes. Proc. ISAAC 2002, LNCS, 2518:44-54, 2002.
Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-coloring and list three-coloring of graphs without induced paths on seven vertices. preprint, 2015.
Flavia Bonomo, Luciano N. Grippo, Martin Milanič, and Martín D. Safe. Graph classes with and without powers of bounded clique-width. Discrete Applied Mathematics, 199:3-15, 2016.
Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free chordal graphs. Proc. MFCS 2015 Part II, LNCS, 9235:139-150, 2015.
Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free split graphs. Discrete Applied Mathematics, (to appear).
Andreas Brandstädt, Joost Engelfriet, Hoàng-Oanh Le, and Vadim V. Lozin. Clique-width for 4-vertex forbidden subgraphs. Theory of Computing Systems, 39(4):561-590, 2006.
Andreas Brandstädt, Vassilis Giakoumakis, and Frédéric Maffray. Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences. Discrete Applied Mathematics, 160(4-5):471-478, 2012.
Andreas Brandstädt, Tilo Klembt, and Suhail Mahfud. P₆- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Mathematics and Theoretical Computer Science, 8(1):173-188, 2006.
Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Gem- and co-gem-free graphs have bounded clique-width. International Journal of Foundations of Computer Science, 15(1):163-185, 2004.
Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Chordal co-gem-free and (P₅,gem)-free graphs have bounded clique-width. Discrete Applied Mathematics, 145(2):232-241, 2005.
Andreas Brandstädt and Suhail Mahfud. Maximum weight stable set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time. Information Processing Letters, 84(5):251-259, 2002.
Hajo Broersma, Petr A. Golovach, Daniël Paulusma, and Jian Song. Determining the chromatic number of triangle-free 2P₃-free graphs in polynomial time. Theoretical Computer Science, 423:1-10, 2012.
Maria Chudnovsky. Coloring graphs with forbidden induced subgraphs. Proc. ICM 2014, IV:291-302, 2014.
Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong. Obstructions for three-coloring graphs with one forbidden induced subgraph. Proc. SODA 2016, pages 1774-1783, 2016.
Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000.
Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring diamond-free graphs. arXiv, 1512.07849, 2015.
Konrad K. Dabrowski, Petr A. Golovach, and Daniël Paulusma. Colouring of graphs with Ramsey-type forbidden subgraphs. Theoretical Computer Science, 522:34-43, 2014.
Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding clique-width via perfect graphs. Proc. LATA 2015, LNCS, 8977:676-688, 2015.
Konrad K. Dabrowski, Vadim V. Lozin, Rajiv Raman, and Bernard Ries. Colouring vertices of triangle-free graphs without forests. Discrete Mathematics, 312(7):1372-1385, 2012.
Konrad K. Dabrowski and Daniël Paulusma. Classifying the clique-width of H-free bipartite graphs. Discrete Applied Mathematics, 200:43-51, 2016.
Konrad K. Dabrowski and Daniël Paulusma. Clique-width of graph classes defined by two forbidden induced subgraphs. The Computer Journal, (in press).
Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. Proc. WG 2001, LNCS, 2204:117-128, 2001.
Jean-Luc Fouquet, Vassilis Giakoumakis, and Jean-Marie Vanherpe. Bipartite graphs totally decomposable by canonical decomposition. International Journal of Foundations of Computer Science, 10(04):513-533, 1999.
Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of colouring graphs with forbidden subgraphs. Journal of Graph Thoery, (in press).
Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984.
Frank Gurski. Graph operations on clique-width bounded graphs. CoRR, abs/cs/0701185, 2007.
Pinar Heggernes, Daniel Meister, and Charis Papadopoulos. Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs. Discrete Applied Mathematics, 160(6):888-901, 2012.
Chính T. Hoàng and D. Adam Lazzarato. Polynomial-time algorithms for minimum weighted colorings of (P₅, ̅P₅)-free graphs and similar graph classes. Discrete Applied Mathematics, 186:106-111, 2015.
Shenwei Huang, Matthew Johnson, and Daniël Paulusma. Narrowing the complexity gap for colouring (C_s,P_t)-free graphs. The Computer Journal, 58(11):3074-3088, 2015.
Marcin Kamiński, Vadim V. Lozin, and Martin Milanič. Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics, 157(12):2747-2761, 2009.
Ton Kloks, Haiko Müller, and Kristina Vušković. Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences. Journal of Combinatorial Theory, Series B, 99(5):733-800, 2009.
Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126(2-3):197-221, 2003.
Daniel Král', Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. Proc. WG 2001, LNCS, 2204:254-262, 2001.
László Lovász. Coverings and coloring of hypergraphs. Congressus Numerantium, VIII:3-12, 1973.
Vadim V. Lozin and Dmitriy S. Malyshev. Vertex coloring of graphs with few obstructions. Discrete Applied Mathematics, (in press).
Vadim V. Lozin and Dieter Rautenbach. On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM Journal on Discrete Mathematics, 18(1):195-206, 2004.
Johann A. Makowsky and Udi Rotics. On the clique-width of graphs with few P₄’s. International Journal of Foundations of Computer Science, 10(03):329-348, 1999.
Dmitriy S. Malyshev. The coloring problem for classes with two small obstructions. Optimization Letters, 8(8):2261-2270, 2014.
Dmitriy S. Malyshev. Two cases of polynomial-time solvability for the coloring problem. Journal of Combinatorial Optimization, 31(2):833-845, 2016.
Sang-Il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms, 5(1):10, 2008.
Bert Randerath and Ingo Schiermeyer. Vertex colouring and forbidden subgraphs - a survey. Graphs and Combinatorics, 20(1):1-40, 2004.
Michaël Rao. MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theoretical Computer Science, 377(1-3):260-267, 2007.
David Schindl. Some new hereditary classes where graph coloring remains NP-hard. Discrete Mathematics, 295(1-3):197-202, 2005.
Alan Tucker. Coloring perfect (K₄-e)-free graphs. Journal of Combinatorial Theory, Series B, 42(3):313-318, 1987.
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Below All Subsets for Some Permutational Counting Problems
We show that the two problems of computing the permanent of an n*n matrix of poly(n)-bit integers and counting the number of Hamiltonian cycles in a directed n-vertex multigraph with exp(poly(n)) edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in o(2^n) time in the worst case. Classic poly(n)2^n time algorithms for the two problems have been known since the early 1960's.
Our algorithms run in 2^{n-Omega(sqrt{n/log(n)})} time.
Matrix Permanent
Hamiltonian Cycles
Asymmetric TSP
17:1-17:11
Regular Paper
Andreas
Björklund
Andreas Björklund
10.4230/LIPIcs.SWAT.2016.17
E. T. Bax. Inclusion and exclusion algorithm for the hamiltonian path problem. Inform. Process. Lett, 47:203-207, 1993.
E. T. Bax and J. Franklin. A permanent algorithm with exp[ω(n ^1/3/2 ln(n))] expected speedup for 0-1 matrices. Algorithmica, 32:157-162, 2002.
R. Bellman. Dynamic programming treatment of the travelling salesman problem. J. Assoc. Comput. Mach, 9:61-63, 1962.
A. Björklund. Counting perfect matchings as fast as ryser. In Proceedings of the ACM-SIAM SODA, pages 914-921, 2012.
A. Björklund. Determinant sums for undirected hamiltonicity. SIAM J. Comput., 43:280-299, 2014.
A. Björklund, H. Dell, and T. Husfeldt. The parity of set systems under random restrictions with applications to exponential time problems. In Proceedings of ICALP,, pages 231-242, 2015.
A. Björklund and T. Husfeldt. The parity of directed hamiltonian cycles. In Proceedings of the IEEE FOCS,, pages 724-735, 2013.
M. Cygan, S. Kratsch, and J. Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the ACM STOC, pages 301-310, 2013.
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H. J. Ryser. Combinatorial Mathematics. The Carus mathematical monographs, The Mathematical Association of America, 1963.
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Extension Complexity, MSO Logic, and Treewidth
We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau.
In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs.
Extension Complexity
FPT
Courcelle's Theorem
MSO Logic
18:1-18:14
Regular Paper
Petr
Kolman
Petr Kolman
Martin
Koutecký
Martin Koutecký
Hans Raj
Tiwary
Hans Raj Tiwary
10.4230/LIPIcs.SWAT.2016.18
S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, June 1991.
D. Avis and H. R. Tiwary. On the extension complexity of combinatorial polytopes. In Proc. ICALP(1), pages 57-68, 2013.
D. Bienstock and G. Munoz. LP approximations to mixed-integer polynomial optimization problems. ArXiv e-prints, January 2015. URL: http://arxiv.org/abs/1501.00288.
http://arxiv.org/abs/1501.00288
H. L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. In Proc. STOC, pages 226-234, 1993.
H. L. Bodlaender. Treewidth: characterizations, applications, and computations. In Proc. of WG, volume 4271 of LNCS, pages 1-14. Springer, 2006.
G. Braun, S. Fiorini, S. Pokutta, and D. Steurer. Approximation limits of linear programs (beyond hierarchies). Math. Oper. Res., 40(3):756-772, 2015.
G. Braun, R. Jain, T. Lee, and S. Pokutta. Information-theoretic approximations of the nonnegative rank. Electronic Colloquium on Computational Complexity, 20:158, 2013.
A. Buchanan and S. Butenko. Tight extended formulations for independent set, 2014. Available on Optimization Online. URL: http://www.optimization-online.org/DB_HTML/2014/09/4540.html.
http://www.optimization-online.org/DB_HTML/2014/09/4540.html
M. Conforti, G. Cornuéjols, and G. Zambelli. Extended formulations in combinatorial optimization. Annals of Operations Research, 204(1):97-143, 2013.
M. Conforti and K. Pashkovich. The projected faces property and polyhedral relations. Mathematical Programming, pages 1-12, 2015.
B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12-75, 1990.
B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In Proc. of WG, volume 1517 of LNCS, pages 125-150. Springer, 1998.
B. Courcelle and M. Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science, 109(1-2):49-82, 1 March 1993.
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S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, and Ronald de Wolf. Exponential lower bounds for polytopes in combinatorial optimization. J. ACM, 62(2):17, 2015.
G. Gottlob, R. Pichler, and F. Wei. Monadic datalog over finite structures with bounded treewidth. In Proc. PODS, pages 165-174, 2007.
B. Grünbaum. Convex Polytopes. Wiley Interscience Publ., London, 1967.
V. Kaibel. Extended formulations in combinatorial optimization. Optima, 85:2-7, 2011.
V. Kaibel and A. Loos. Branched polyhedral systems. In Proc. IPCO, volume 6080 of LNCS, pages 177-190. Springer, 2010.
V. Kaibel and K. Pashkovich. Constructing extended formulations from reflection relations. In Proc. IPCO, volume 6655 of LNCS, pages 287-300. Springer, 2011.
L. Kaiser, M. Lang, S. Leßenich, and Ch. Löding. A Unified Approach to Boundedness Properties in MSO. In Proc. of CSL, volume 41 of LIPIcs, pages 441-456, 2015.
T. Kloks. Treewidth: Computations and Approximations, volume 842 of LNCS. Springer, 1994.
J. Kneis, A. Langer, and P. Rossmanith. Courcelle’s theorem - A game-theoretic approach. Discrete Optimization, 8(4):568-594, 2011.
P. G. Kolaitis and M. Y. Vardi. Conjunctive-query containment and constraint satisfaction. In Proc. PODS, 1998.
P. Kolman and M. Koutecký. Extended formulation for CSP that is compact for instances of bounded treewidth. Electr. J. Comb., 22(4):P4.30, 2015.
S. Kreutzer. Algorithmic meta-theorems. In Proc. of IWPEC, volume 5018 of LNCS, pages 10-12. Springer, 2008.
A. Langer, F. Reidl, P. Rossmanith, and S. Sikdar. Practical algorithms for MSO model-checking on tree-decomposable graphs. Computer Science Review, 13-14:39-74, 2014.
M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157-270. Springer, 2009.
J. R. Lee, P. Raghavendra, and D. Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proc. STOC, pages 567-576, 2015.
L. Libkin. Elements of Finite Model Theory. Springer, Berlin, 2004.
F. Margot. Composition de polytopes combinatoires: une approche par projection. PhD thesis, École polytechnique fédérale de Lausanne, 1994.
R. K. Martin, R. L. Rardin, and B. A. Campbell. Polyhedral characterization of discrete dynamic programming. Oper. Res., 38(1):127-138, February 1990.
M. Sellmann. The polytope of tree-structured binary constraint satisfaction problems. In Proc. CPAIOR, volume 5015 of LNCS, pages 367-371. Springer, 2008.
M. Sellmann, L. Mercier, and D. H. Leventhal. The linear programming polytope of binary constraint problems with bounded tree-width. In Proc. CPAIOR, volume 4510 of LNCS, pages 275-287. Springer, 2007.
F. Vanderbeck and L. A. Wolsey. Reformulation and decomposition of integer programs. In 50 Years of Integer Programming 1958-2008, pages 431-502. Springer, 2010.
L. A. Wolsey. Using extended formulations in practice. Optima, 85:7-9, 2011.
M. Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441-466, 1991.
G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer, 1995.
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Optimal Online Escape Path Against a Certificate
More than fifty years ago Bellman asked for the best escape path within a known forest but for an unknown starting position. This deterministic finite path is the shortest path that leads out of a given environment from any starting point. There are some worst case positions where the full path length is required. Up to now such a fixed ultimate optimal escape path for a known shape for any starting position is only known for some special convex shapes (i.e., circles, strips of a given width, fat convex bodies, some isosceles triangles).
Therefore, we introduce a different, simple and intuitive escape path, the so-called certificate path which make use of some additional information w.r.t. the starting point s. This escape path depends on the starting position s and takes the distances from s to the outer boundary of the environment into account. Because of this, in the above convex examples the certificate path always (for any position s) leaves the environment earlier than the ultimate escape path.
Next we assume that neither the precise shape of the environment nor the location of the starting point is not known, we have much less information. For a class of environments (convex shapes and shapes with kernel positions) we design an online strategy that always leaves the environment. We show that the path length for leaving the environment is always shorter than 3.318764 the length of the corresponding certificate path. We also give a lower bound of 3.313126 which shows that for the above class of environments the factor 3.318764 is (almost) tight.
Search games
online algorithms
escape path
competitive analysis
spiral conjecture
19:1-19:14
Regular Paper
Elmar
Langetepe
Elmar Langetepe
David
Kübel
David Kübel
10.4230/LIPIcs.SWAT.2016.19
Steve Alpern and Shmuel Gal. The Theory of Search Games and Rendezvous. Kluwer Academic Publications, 2003.
R. Baeza-Yates, J. Culberson, and G. Rawlins. Searching in the plane. Inform. Comput., 106:234-252, 1993.
Richard Bellman. Minimization problem. Bull. Amer. Math. Soc., 62(3):270, 1956.
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P. Coulton and Y. Movshovich. Besicovitch triangles cover unit arcs. Geometriae Dedicata, 123(1):79-88, 2006. URL: http://dx.doi.org/10.1007/s10711-006-9107-7.
http://dx.doi.org/10.1007/s10711-006-9107-7
Andrea Eubeler, Rudolf Fleischer, Tom Kamphans, Rolf Klein, Elmar Langetepe, and Gerhard Trippen. Competitive online searching for a ray in the plane. In Sándor Fekete, Rudolf Fleischer, Rolf Klein, and Alejandro L\a'opez-Ortiz, editors, Robot Navigation, number 06421 in Dagstuhl Seminar Proceedings, 2006.
Amos Fiat and Gerhard Woeginger, editors. On-line Algorithms: The State of the Art, volume 1442 of Lecture Notes Comput. Sci. Springer-Verlag, 1998.
Steven R. Finch. The logarithmic spiral conjecture, 2005.
Steven R. Finch and John E. Wetzel. Lost in a forest. The American Mathematical Monthly, 111(8):pp. 645-654, 2004.
Steven R. Finch and Li-Yan Zhu. Searching for a shoreline. arXiv:math/0501123v1, 2005.
Rudolf Fleischer, Tom Kamphans, Rolf Klein, Elmar Langetepe, and Gerhard Trippen. Competitive online approximation of the optimal search ratio. Siam J. Comput., pages 881-898, 2008.
S. Gal and D. Chazan. On the optimality of the exponential functions for some minmax problems. SIAM J. Appl. Math., 30:324-348, 1976.
Shmuel Gal. Search Games, volume 149 of Mathematics in Science and Engeneering. Academic Press, New York, 1980.
Brian Gluss. The minimax path in a search for a circle in a plane. Naval Research Logistics Quarterly, 8(4):357-360, 1961. URL: http://dx.doi.org/10.1002/nav.3800080404.
http://dx.doi.org/10.1002/nav.3800080404
Christian Icking, Thomas Kamphans, Rolf Klein, and Elmar Langetepe. On the competitive complexity of navigation tasks. In H. Bunke and et al., editors, Sensor Based Intelligent Robots, volume 2238 of LNCS, pages 245-258. Springer, 2002.
David Kirkpatrick. Hyperbolic dovetailing. In Amos Fiat and Peter Sanders, editors, Algorithms - ESA 2009, volume 5757 of Lecture Notes in Computer Science, pages 516-527. Springer Berlin Heidelberg, 2009. URL: http://dx.doi.org/10.1007/978-3-642-04128-0_46.
http://dx.doi.org/10.1007/978-3-642-04128-0_46
David Kirkpatrick. Personal communication, 2015. Workshop on Geometric Problems on Sensor Networks and Robots, Supported by NSF grant CCF 1017539, Organized by Peter Brass (CCNY) and Jon Lenchner (IBM Research).
Elias Koutsoupias, Christos H. Papadimitriou, and Mihalis Yannakakis. Searching a fixed graph. In Proc. 23th Internat. Colloq. Automata Lang. Program., volume 1099 of Lecture Notes Comput. Sci., pages 280-289. Springer, 1996.
Elmar Langetepe. On the optimality of spiral search. In SODA 2010: Proc. 21st Annu. ACM-SIAM Symp. Disc. Algor., pages 1-12, 2010.
Elmar Langetepe and David Kübel. Optimal online escape path against a certificate. Technical Report arXiv:1604.05972, University of Bonn, 2016.
N. S. V. Rao, S. Kareti, W. Shi, and S. S. Iyengar. Robot navigation in unknown terrains: introductory survey of non-heuristic algorithms. Technical Report ORNL/TM-12410, Oak Ridge National Laboratory, 1993.
S. Schuierer. Lower bounds in on-line geometric searching. Comput. Geom. Theory Appl., 18:37-53, 2001.
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Lagrangian Duality based Algorithms in Online Energy-Efficient Scheduling
We study online scheduling problems in the general energy model of speed scaling with power down. The latter is a combination of the two extensively studied energy models, speed scaling and power down, toward a more realistic one. Due to the limits of the current techniques, only few results have been known in the general energy model in contrast to the large literature of the previous ones.
In the paper, we consider a Lagrangian duality based approach to design and analyze algorithms in the general energy model. We show the applicability of the approach to problems which are unlikely to admit a convex relaxation. Specifically, we consider the problem of minimizing energy with a single machine in which jobs arrive online and have to be processed before their deadlines. We present an alpha^alpha-competitive algorithm (whose the analysis is tight up to a constant factor) where the energy power function is of typical form z^alpha + g for constants alpha > 2 and g non-negative. Besides, we also consider the problem of minimizing the weighted flow-time plus energy. We give an O(alpha/ln(alpha))-competitive algorithm; that matches (up to a constant factor) to the currently best known algorithm for this problem in the restricted model of speed scaling.
Online Scheduling
Energy Minimization
Speed Scaling and Power-down
Lagrangian Duality
20:1-20:14
Regular Paper
Nguyen
Kim Thang
Nguyen Kim Thang
10.4230/LIPIcs.SWAT.2016.20
Susanne Albers. Energy-efficient algorithms. Commun. ACM, 53(5):86-96, 2010.
Susanne Albers and Antonios Antoniadis. Race to idle: new algorithms for speed scaling with a sleep state. ACM Transactions on Algorithms (TALG), 10(2):9, 2014.
S. Anand, Naveen Garg, and Amit Kumar. Resource augmentation for weighted flow-time explained by dual fitting. In Proc. 23rd ACM-SIAM Symposium on Discrete Algorithms, pages 1228-1241, 2012.
Spyros Angelopoulos, Giorgio Lucarelli, and Kim Thang Nguyen. Primal-dual and dual-fitting analysis of online scheduling algorithms for generalized flow time problems. In Proc. 23rd European Symposium of Algorithms (ESA), pages 35-46. Springer, 2015.
Antonios Antoniadis, Chien-Chung Huang, and Sebastian Ott. A fully polynomial-time approximation scheme for speed scaling with sleep state. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1102-1113. SIAM, 2015.
Yossi Azar, Nikhil R. Devanur, Zhiyi Huang, and Debmalya Panigrahi. Speed scaling in the non-clairvoyant model. In Proc. 27th ACM on Symposium on Parallelism in Algorithms and Architectures, pages 133-142, 2015.
Evripidis Bampis, Christoph Dürr, Fadi Kacem, and Ioannis Milis. Speed scaling with power down scheduling for agreeable deadlines. Sustainable Computing: Informatics and Systems, 2(4):184-189, 2012.
Nikhil Bansal, Ho-Leung Chan, Dmitriy Katz, and Kirk Pruhs. Improved bounds for speed scaling in devices obeying the cube-root rule. Theory of Computing, 8(1):209-229, 2012.
Nikhil Bansal, Ho-Leung Chan, and Kirk Pruhs. Speed scaling with an arbitrary power function. In Proc. 20th ACM-SIAM Symposium on Discrete Algorithms, pages 693-701, 2009.
Nikhil Bansal, Tracy Kimbrel, and Kirk Pruhs. Speed scaling to manage energy and temperature. J. ACM, 54(1), 2007.
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.
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.
Xin Han, Tak Wah Lam, Lap-Kei Lee, Isaac Kar-Keung To, and Prudence W. H. Wong. Deadline scheduling and power management for speed bounded processors. Theor. Comput. Sci., 411(40-42):3587-3600, 2010.
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, Kamesh Munagala, and Kirk Pruhs. Selfishmigrate: A scalable algorithm for non-clairvoyantly scheduling heterogeneous processors. In Proc. 55th IEEE Symposium on Foundations of Computer Science, 2014.
Sandy Irani, Sandeep K. Shukla, and Rajesh Gupta. Algorithms for power savings. ACM Transactions on Algorithms, 3(4), 2007.
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.
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Online Dominating Set
This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. In most cases, we obtain tight bounds on the competitive ratios. Our results show that requiring the graphs to be presented in a connected fashion allows the online algorithms to obtain provably better solutions. Furthermore, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm's disadvantage.
online algorithms
dominating set
competitive analysis
graph classes
connected graphs
21:1-21:15
Regular Paper
Joan
Boyar
Joan Boyar
Stephan J.
Eidenbenz
Stephan J. Eidenbenz
Lene M.
Favrholdt
Lene M. Favrholdt
Michal
Kotrbcik
Michal Kotrbcik
Kim S.
Larsen
Kim S. Larsen
10.4230/LIPIcs.SWAT.2016.21
J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks, and R. Niedermeier. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica, 33(4):461-493, 2002.
B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153-180, 1994.
C. Berge. Theory of Graphs and its Applications. Meuthen, London, 1962.
M. Böhm, J. Sgall, and P. Veselý. Online colored bin packing. In E. Bampis and O. Svensson, editors, 12th International Workshop on Approximation and Online Algorithms (WAOA), volume 8952 of Lecture Notes in Computer Science, pages 35-46. Springer, 2015.
A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998.
J. Boyar, S. J. Eidenbenz, L. M. Favrholdt, M. Kotrbčík, and K. S. Larsen. Online dominating set. Technical Report arXiv:1604.05172 [cs.DS], arXiv, 2016.
J. Boyar and K. S. Larsen. The seat reservation problem. Algorithmica, 25(4):403-417, 1999.
M. Chrobak, J. Sgall, and G. J. Woeginger. Two-bounded-space bin packing revisited. In C. Demetrescu and M. M. Halldórsson, editors, 19th Annual European Symposium (ESA), volume 6942 of Lecture Notes in Computer Science, pages 263-274. Springer, 2011.
V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979.
B. Das and V. Bharghavan. Routing in ad-hoc networks using minimum connected dominating sets. In IEEE International Conference on Communications (ICC), volume 1, pages 376-380, 1997.
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D.-Z. Du and P.-J. Wan. Connected Dominating Set: Theory and Applications. Springer, New York, 2013.
U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4):634-652, 1998.
T. W. Haynes, S. Hedetniemi, and P. Slater. Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998.
M. Henning and A. Yao. Total Domination in Graphs. Springer, New York, 2013.
A. R. Karlin, M. S. Manasse, L. Rudolph, and D. D. Sleator. Competitive snoopy caching. Algorithmica, 3:79-119, 1988.
R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972.
G.-H. King and W.-G. Tzeng. On-line algorithms for the dominating set problem. Information Processing Letters, 61(1):11-14, 1997.
D. König. Theorie der Endlichen und Unendlichen Graphen. Chelsea, New York, 1950.
C. L. Liu. Introduction to Combinatorial Mathematics. McGraw-Hill, New York, 1968.
O. Ore. Theory of Graphs, volume 38 of Colloquium Publications. American Mathematical Society, Providence, 1962.
D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202-208, 1985.
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Sorting Under Forbidden Comparisons
In this paper we study the problem of sorting under forbidden comparisons where some pairs of elements may not be compared (forbidden pairs). Along with the set of elements V the input to our problem is a graph G(V, E), whose edges represents the pairs that we can compare in constant time. Given a graph with n vertices and m = binom(n)(2) - q edges we propose the first non-trivial deterministic algorithm which makes O((q + n)*log(n)) comparisons with a total complexity of O(n^2 + q^(omega/2)), where omega is the exponent in the complexity of matrix multiplication. We also propose a simple randomized algorithm for the problem which makes widetilde O(n^2/sqrt(q+n) + nsqrt(q)) probes with high probability. When the input graph is random we show that widetilde O(min(n^(3/2), pn^2)) probes suffice, where p is the edge probability.
Sorting
Random Graphs
Complexity
22:1-22:13
Regular Paper
Indranil
Banerjee
Indranil Banerjee
Dana
Richards
Dana Richards
10.4230/LIPIcs.SWAT.2016.22
Miklós Ajtai, János Komlós, William Steiger, and Endre Szemerédi. Almost sorting in one round. Randomness and Computation, 5:117-125, 1989.
Mohamad Akra and Louay Bazzi. On the solution of linear recurrence equations. Comp. Opt. and Appl., 10(2):195-210, 1998. URL: http://dx.doi.org/10.1023/A:1018373005182,
http://dx.doi.org/10.1023/A:1018373005182
Noga Alon, Manuel Blum, Amos Fiat, Sampath Kannan, Moni Naor, and Rafail Ostrovsky. Matching nuts and bolts. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. 23-25 January 1994, Arlington, Virginia., pages 690-696, 1994. URL: http://dl.acm.org/citation.cfm?id=314464.314673.
http://dl.acm.org/citation.cfm?id=314464.314673
Stanislav Angelov, Keshav Kunal, and Andrew McGregor. Sorting and selection with random costs. In LATIN 2008: Theoretical Informatics, 8th Latin American Symposium, Búzios, Brazil, April 7-11, 2008, Proceedings, pages 48-59, 2008. URL: http://dx.doi.org/10.1007/978-3-540-78773-0_5,
http://dx.doi.org/10.1007/978-3-540-78773-0_5
Béla Bollobás and Graham Brightwell. Graphs whose every transitive orientation contains almost every relation. Israel Journal of Mathematics, 59(1):112-128, 1987.
Béla Bollobás and Graham R. Brightwell. Transitive orientations of graphs. SIAM J. Comput., 17(6):1119-1133, 1988. URL: http://dx.doi.org/10.1137/0217072,
http://dx.doi.org/10.1137/0217072
Béla Bollobás and Moshe Rosenfeld. Sorting in one round. Israel Journal of Mathematics, 38(1-2):154-160, 1981.
Jean Cardinal and Samuel Fiorini. On generalized comparison-based sorting problems. In Space-Efficient Data Structures, Streams, and Algorithms - Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday, pages 164-175, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40273-9_12,
http://dx.doi.org/10.1007/978-3-642-40273-9_12
Jean Cardinal, Samuel Fiorini, Gwenaël Joret, Raphaël M. Jungers, and J. Ian Munro. An efficient algorithm for partial order production. SIAM J. Comput., 39(7):2927-2940, 2010. URL: http://dx.doi.org/10.1137/090759860,
http://dx.doi.org/10.1137/090759860
Moses Charikar, Ronald Fagin, Venkatesan Guruswami, Jon M. Kleinberg, Prabhakar Raghavan, and Amit Sahai. Query strategies for priced information. J. Comput. Syst. Sci., 64(4):785-819, 2002. URL: http://dx.doi.org/10.1006/jcss.2002.1828,
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SD Chatterji. The number of topologies on n points, kent state university. NASA Technical Report, 1966.
Constantinos Daskalakis, Richard M. Karp, Elchanan Mossel, Samantha Riesenfeld, and Elad Verbin. Sorting and selection in posets. SIAM J. Comput., 40(3):597-622, 2011. URL: http://dx.doi.org/10.1137/070697720,
http://dx.doi.org/10.1137/070697720
Martin E. Dyer, Alan M. Frieze, and Ravi Kannan. A random polynomial time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1-17, 1991. URL: http://doi.acm.org/10.1145/102782.102783, URL: http://dx.doi.org/10.1145/102782.102783.
http://dx.doi.org/10.1145/102782.102783
Ulrich Faigle and György Turán. Sorting and recognition problems for ordered sets. SIAM J. Comput., 17(1):100-113, 1988. URL: http://dx.doi.org/10.1137/0217007,
http://dx.doi.org/10.1137/0217007
Wayne Goddard, Claire Kenyon, Valerie King, and Leonard J. Schulman. Optimal randomized algorithms for local sorting and set-maxima. SIAM J. Comput., 22(2):272-283, 1993. URL: http://dx.doi.org/10.1137/0222020,
http://dx.doi.org/10.1137/0222020
Anupam Gupta and Amit Kumar. Sorting and selection with structured costs. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 416-425, 2001. URL: http://dx.doi.org/10.1109/SFCS.2001.959916,
http://dx.doi.org/10.1109/SFCS.2001.959916
Zhiyi Huang, Sampath Kannan, and Sanjeev Khanna. Algorithms for the generalized sorting problem. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 738-747, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.54,
http://dx.doi.org/10.1109/FOCS.2011.54
Nabil Kahale and Leonard J. Schulman. Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph. Combinatorica, 16(3):383-397, 1996. URL: http://dx.doi.org/10.1007/BF01261322,
http://dx.doi.org/10.1007/BF01261322
Jeff Kahn and Michael Saks. Balancing poset extensions. Order, 1(2):113-126, 1984.
János Komlós, Yuan Ma, and Endre Szemerédi. Matching nuts and bolts in o(n log n) time. SIAM J. Discrete Math., 11(3):347-372, 1998. URL: http://dx.doi.org/10.1137/S0895480196304982,
http://dx.doi.org/10.1137/S0895480196304982
John E. Savage. Models of computation - exploring the power of computing. Addison-Wesley, 1998.
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Total Stability in Stable Matching Games
The stable marriage problem (SMP) can be seen as a typical game, where each player wants to obtain the best possible partner by manipulating his/her preference list. Thus the set Q of preference lists submitted to the matching agency may differ from P, the set of true preference lists. In this paper, we study the stability of the stated lists in Q. If Q is not Nash equilibrium, i.e., if a player can obtain a strictly better partner (with respect to the preference order in P) by only changing his/her list, then in the view of standard game theory, Q is vulnerable. In the case of SMP, however, we need to consider another factor, namely that all valid matchings should not include any "blocking pairs" with respect to P. Thus, if the above manipulation of a player introduces blocking pairs, it would prevent this manipulation. Consequently, we say Q is totally stable if either Q is a Nash equilibrium or if any attempt at manipulation by a single player causes blocking pairs with respect to P. We study the complexity of testing the total stability of a stated strategy. It is known that this question is answered in polynomial time if the instance (P,Q) always satisfies P=Q. We extend this polynomially solvable class to the general one, where P and Q may be arbitrarily different.
stable matching
Gale-Shapley algorithm
manipulation
stability
Nash equilibrium
23:1-23:12
Regular Paper
Sushmita
Gupta
Sushmita Gupta
Kazuo
Iwama
Kazuo Iwama
Shuichi
Miyazaki
Shuichi Miyazaki
10.4230/LIPIcs.SWAT.2016.23
G. Demange, D. Gale, and M. Sotomayor. A further remark on the stable matching problem. Discrete Applied Mathematics, 16:217-222, 1987.
L. E. Dubins and D. A. Freedman. Machiavelli and the Gale-Shapley algorithm. The American Mathematical Monthly, 88(7):485-494, 1981.
L. Ehlers. Truncation strategies in matching markets. Mathematics of Operations Research, 33(2):327-335, 2008.
D. Gale and L. S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69:9-15, 1962.
D. Gale and M. Sotomayor. Ms. Machiavelli and the Gale-Shapley algorithm. American Mathematical Monthly, 92(4):261-268, 1985.
D. Gusfield and R. W. Irving. The Stable Marriage Problem-Structure and Algorithm. The MIT Press, 1989.
N. Immorlica and M. Mahidian. Marriage, honesty and stability. In Proceedings of SODA'05, pages 53-62, 2005.
H. Kobayashi and T. Matsui. Cheating strategies for the Gale-Shapley algorithm with complete preference lists. Algorithmica, 58:151-169, 2010.
M. S. Piny, F. Rossi, K. B. Veneble, and T. Walsh. Manipulation complexity and gender neutrality in in stable marriage procedures. Auton. Agent Multi-Agent Systems, 22(183-199), 2011.
A. E. Roth. Misrepresentation and stability in the marriage problem. Journal of Economic Theory, 34:383-387, 1984.
A. E. Roth and U. G. Rothblum. Truncation strategies in matching markets-in search of advice for participants. Econometrica, 67(1):21-43, 1999.
A. E. Roth and M. Sotomayor. Two-Sided Matching: A Study in Game Theoretic Modeling and Analysis. Cambridge Univ. Press, 1990.
C-P. Teo, J. Sethuraman, and W-P. Tan. Gale-Shapley stable marriage problem revisited: Strategic issues and applications. Management Science, 47(9):1252-1267, 2001.
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Estimating The Makespan of The Two-Valued Restricted Assignment Problem
We consider a special case of the scheduling problem on unrelated machines, namely the Restricted Assignment Problem with two different processing times. We show that the configuration LP has an integrality gap of at most 5/3 ~ 1.667 for this problem. This allows us to estimate the optimal makespan within a factor of 5/3, improving upon the previously best known estimation algorithm with ratio 11/6 ~ 1.833 due to Chakrabarty, Khanna, and Li.
unrelated scheduling
restricted assignment
configuration LP
integrality gap
estimation algorithm
24:1-24:13
Regular Paper
Klaus
Jansen
Klaus Jansen
Kati
Land
Kati Land
Marten
Maack
Marten Maack
10.4230/LIPIcs.SWAT.2016.24
Nikhil Bansal and Maxim Sviridenko. The santa claus problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, (STOC 2006), pages 31-40, 2006.
Deeparnab Chakrabarty, Sanjeev Khanna, and Shi Li. On (1, epsilon)-restricted assignment makespan minimization. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 1087-1101, 2015.
Tomáš Ebenlendr, Marek Krčál, and Jiří Sgall. Graph balancing: A special case of scheduling unrelated parallel machines. Algorithmica, 68(1):62-80, 2014.
Chien-Chung Huang and Sebastian Ott. A combinatorial approximation algorithm for graph balancing with light hyper edges. CoRR, abs/1507.07396, 2015.
Stavros G Kolliopoulos and Yannis Moysoglou. The 2-valued case of makespan minimization with assignment constraints. Information Processing Letters, 113(1):39-43, 2013.
Jan Karel Lenstra, David B Shmoys, and Éva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical programming, 46(1-3):259-271, 1990.
Evgeny V. Shchepin and Nodari Vakhania. An optimal rounding gives a better approximation for scheduling unrelated machines. Operations Research Letters, 33(2):127-133, 2005.
David B Shmoys and Éva Tardos. An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62(1-3):461-474, 1993.
Ola Svensson. Santa claus schedules jobs on unrelated machines. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC 2011), pages 617-626, 2011.
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A Plane 1.88-Spanner for Points in Convex Position
Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).
points in convex position
plane spanner
25:1-25:14
Regular Paper
Mahdi
Amani
Mahdi Amani
Ahmad
Biniaz
Ahmad Biniaz
Prosenjit
Bose
Prosenjit Bose
Jean-Lou
De Carufel
Jean-Lou De Carufel
Anil
Maheshwari
Anil Maheshwari
Michiel
Smid
Michiel Smid
10.4230/LIPIcs.SWAT.2016.25
R. V. Benson. Euclidean geometry and convexity. McGraw-Hill, 1966.
A. L. Cauchy. Note sur divers théorèmes relatifs á la rectification des courbes et á la quadrature des surfaces. C. R. Acad. Sci. Paris, 13:1060-1065, 1841.
T. M. Chan. A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM, 57(3), 2010.
L. P. Chew. There is a planar graph almost as good as the complete graph. In Proceedings of the 2nd ACM Symposium on Computational Geometry, pages 169-177, 1986.
L. P. Chew. There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences, 39:205-219, 1989.
S. Cui, I. A. Kanj, and G. Xia. On the stretch factor of Delaunay triangulations of points in convex position. Computational Geometry: Theory and Applications, 44:104-109, 2011.
D. P. Dobkin, S. J. Friedman, and K. J. Supowit. Delaunay graphs are almost as good as complete graphs. Discrete &Computational Geometry, 5:399-407, 1990.
A. Dumitrescu and A. Ghosh. Lower bounds on the dilation of plane spanners. In Proceedings of the 2nd International Conference on Algorithms and Discrete Applied Mathematics, CALDAM, pages 139-151, 2016.
R. Janardan. On maintaining the width and diameter of a planar point-set online. Int. J. Comput. Geometry Appl., 3(3):331-344, 1993.
J. M. Keil and C. A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discrete &Computational Geometry, 7:13-28, 1992.
W. Mulzer. Minimum dilation triangulations for the regular n-gon. Master’s thesis, Freie Universität Berlin, Germany, 2004.
G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, Cambridge, UK, 2007.
G. Xia. The stretch factor of the Delaunay triangulation is less than 1.998. SIAM Journal on Computing, 42:1620-1659, 2013.
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Approximating the Integral Fréchet Distance
We present a pseudo-polynomial time (1 + epsilon)-approximation algorithm for computing the integral and average Fréchet distance between two given polygonal curves T_1 and T_2. The running time is in O(zeta^{4}n^4/epsilon^2) where n is the complexity of T_1 and T_2 and zeta is the maximal ratio of the lengths of any pair of segments from T_1 and T_2.
Furthermore, we give relations between weighted shortest paths inside a single parameter cell C and the monotone free space axis of C. As a result we present a simple construction of weighted shortest paths inside a parameter cell. Additionally, such a shortest path provides an optimal solution for the partial Fréchet similarity of segments for all leash lengths. These two aspects are related to each other and are of independent interest.
Fréchet distance
partial Fréchet similarity
curve matching
26:1-26:14
Regular Paper
Anil
Maheshwari
Anil Maheshwari
Jörg-Rüdiger
Sack
Jörg-Rüdiger Sack
Christian
Scheffer
Christian Scheffer
10.4230/LIPIcs.SWAT.2016.26
Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75-91, 1995. URL: http://dx.doi.org/10.1142/S0218195995000064.
http://dx.doi.org/10.1142/S0218195995000064
Kevin Buchin, Maike Buchin, Wouter Meulemans, and Bettina Speckmann. Locally correct Fréchet matchings. In Leah Epstein and Paolo Ferragina, editors, ESA, volume 7501 of Lecture Notes in Computer Science, pages 229-240. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33090-2_21.
http://dx.doi.org/10.1007/978-3-642-33090-2_21
Kevin Buchin, Maike Buchin, and Yusu Wang. Exact algorithms for partial curve matching via the Fréchet distance. In Claire Mathieu, editor, SODA, pages 645-654. SIAM, 2009. URL: http://dx.doi.org/10.1137/1.9781611973068.
http://dx.doi.org/10.1137/1.9781611973068
Maike Buchin. On the computability of the Fréchet distance between triangulated surfaces. Ph.D. thesis, Dept. of Comput. Sci., Freie Universität, Berlin, 2007.
Jean-Lou De Carufel, Amin Gheibi, Anil Maheshwari, Jörg-Rüdiger Sack, and Christian Scheffer. Similarity of polygonal curves in the presence of outliers. Comput. Geom., 47(5):625-641, 2014.
Isabel F. Cruz, Craig A. Knoblock, Peer Kröger, Egemen Tanin, and Peter Widmayer, editors. SIGSPATIAL 2012 International Conference on Advances in Geographic Information Systems (formerly known as GIS), SIGSPATIAL'12, Redondo Beach, CA, USA, November 7-9, 2012. ACM, 2012.
Alon Efrat, Quanfu Fan, and Suresh Venkatasubramanian. Curve matching, time warping, and light fields: New algorithms for computing similarity between curves. Journal of Mathematical Imaging and Vision, 27(3):203-216, 2007. URL: http://dx.doi.org/10.1007/s10851-006-0647-0.
http://dx.doi.org/10.1007/s10851-006-0647-0
Stefan Funke and Edgar A. Ramos. Smooth-surface reconstruction in near-linear time. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA., pages 781-790, 2002.
Joachim Gudmundsson, Patrick Laube, and Thomas Wolle. Movement patterns in spatio-temporal data. In Shashi Shekhar and Hui Xiong, editors, Encyclopedia of GIS, pages 726-732. Springer, 2008. URL: http://dx.doi.org/10.1007/978-0-387-35973-1_823.
http://dx.doi.org/10.1007/978-0-387-35973-1_823
Joachim Gudmundsson and Nacho Valladares. A GPU approach to subtrajectory clustering using the Fréchet distance. In Cruz et al. [6], pages 259-268. URL: http://dx.doi.org/10.1145/2424321.2424355.
http://dx.doi.org/10.1145/2424321.2424355
Joachim Gudmundsson and Thomas Wolle. Football analysis using spatio-temporal tools. In Cruz et al. [6], pages 566-569. URL: http://dx.doi.org/10.1145/2424321.2424417.
http://dx.doi.org/10.1145/2424321.2424417
Lawrence R. Rabiner and Biing-Hwang Juang. Fundamentals of speech recognition. Prentice Hall Signal Processing series. Prentice Hall, 1993.
Günter Rote. Lexicographic fréechet matchings. In Proceedings of the 32st European Workshop on Computational Geometry, pages 101-104, 2014.
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A Clustering-Based Approach to Kinetic Closest Pair
Given a set P of n moving points in fixed dimension d, where the trajectory of each point is a polynomial of degree bounded by some constant, we present a kinetic data structure (KDS) for maintenance of the closest pair on P.
Assuming the closest pair distance is between 1 and Delta over time, our KDS uses O(n log Delta) space and processes O(n^2 beta log Delta log n + n^2 beta log Delta log log Delta)) events, each in worst-case time O(log^2 n + log^2 log Delta). Here, beta is an extremely slow-growing function. The locality of the KDS is O(log n + log log Delta). Our closest pair KDS supports insertions and deletions of points. An insertion or deletion takes worst-case time O(log Delta log^2 n + log Delta log^2 log Delta).
Also, we use a similar approach to provide a KDS for the all epsilon-nearest neighbors in R^d.
The complexities of the previous KDSs, for both closest pair and all epsilon-nearest neighbors, have polylogarithmic factor, where the number of logs depends on dimension d. Assuming Delta is polynomial in n, our KDSs obtain improvements on the previous KDSs.
Our solutions are based on a kinetic clustering on P. Though we use ideas from the previous clustering KDS by Hershberger, we simplify and improve his work.
Kinetic Data Structure
Clustering
Closest Pair
All Nearest Neighbors
28:1-28:13
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Zahed
Rahmati
Zahed Rahmati
10.4230/LIPIcs.SWAT.2016.28
Pankaj K. Agarwal, Haim Kaplan, and Micha Sharir. Kinetic and dynamic data structures for closest pair and all nearest neighbors. ACM Transactions on Algorithms, 5:4:1-37, 2008.
Julien Basch, Leonidas J. Guibas, and John Hershberger. Data structures for mobile data. Journal of Algorithms, 31:1-19, 1999.
Julien Basch, Leonidas J. Guibas, and Li Zhang. Proximity problems on moving points. In Proceedings of the 13th Annual Symposium on Computational Geometry (SoCG'97), pages 344-351, New York, NY, USA, 1997. ACM.
H. Brönnimann and M.T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete &Computational Geometry, 14(1):463-479, 1995.
Timothy M. Chan and Zahed Rahmati. Approximating the minimum closest pair distance and nearest neighbor distances of linearly moving points. Computational Geometry, 2016.
Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing (STOC'88), pages 434-444, New York, NY, USA, 1988. ACM.
Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133-137, 1981.
Jie Gao, Leonidas Guibas, John Hershberger, Li Zhang, and An Zhu. Discrete mobile centers. Discrete &Computational Geometry, 30(1):45-63, 2003.
Teofilo F. Gonzalez. Covering a set of points in multidimensional space. Information Processing Letters, 40(4):181-188, 1991.
John Hershberger. Smooth kinetic maintenance of clusters. Computational Geometry, 31(1–2):3-30, 2005.
Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985.
Zahed Rahmati. Simple, Faster Kinetic Data Structures. PhD thesis, University of Victoria, 2014.
Zahed Rahmati, Mohammad Ali Abam, Valerie King, and Sue Whitesides. Kinetic k-semi-Yao graph and its applications. Computational Geometry, 2016.
Zahed Rahmati, Mohammad Ali Abam, Valerie King, Sue Whitesides, and Alireza Zarei. A simple, faster method for kinetic proximity problems. Computational Geometry, 48(4):342-359, 2015.
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Constrained Geodesic Centers of a Simple Polygon
For any two points in a simple polygon P, the geodesic distance between them is the length of the shortest path contained in P that connects them. A geodesic center of a set S of sites (points) with respect to P is a point in P that minimizes the geodesic distance to its farthest site. In many realistic facility location problems, however, the facilities are constrained to lie in feasible regions. In this paper, we show how to compute the geodesic centers constrained to a set of line segments or simple polygonal regions contained in P. Our results provide substantial improvements over previous algorithms.
Geodesic distance
simple polygons
constrained center
facility location
farthest-point Voronoi diagram
29:1-29:13
Regular Paper
Eunjin
Oh
Eunjin Oh
Wanbin
Son
Wanbin Son
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SWAT.2016.29
Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A linear-time algorithm for the geodesic center of a simple polygon. In Proc. 31st Int'l Symposium on Computational Geometry (SoCG 2015), pages 209-223, 2015.
Boris Aronov, Steven Fortune, and Gordon Wilfong. The furthest-site geodesic Voronoi diagram. Discrete &Computational Geometry, 9(1):217-255, 1993.
Tetsuo Asano and Godfried Toussaint. Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University, 1985.
Luis Barba. Disk constrained 1-center queries. In Proc. 24th Canadian Conference on Computational Geometry (CCCG 2012), pages 15-19, 2012.
Luis Barba, Prosenjit Bose, and Stefan Langerman. Optimal algorithms for constrained 1-center problems. In Proc. 11th Latin American Theoretical Informatics Symposium (LATIN 2014), pages 84-95, 2014.
Prosenjit Bose, Stefan Langerman, and Sasanka Roy. Smallest enclosing circle centered on a query line segment. In Proc. 20th Canadian Conference on Computational Geometry (CCCG 2008), pages 167-170, 2008.
Prosenjit Bose and Godfried Toussaint. Computing the constrained Euclidean geodesic and link center of a simple polygon with applications. In Proc. 14th Computer Graphics International (CGI 1996), pages 102-110, 1996.
Prosenjit Bose and Qingda Wang. Facility location constrained to a polygonal domain. In Proc. 5th Latin American Theoretical Informatics Symp. (LATIN'02), pages 153-164, 2002.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete &Computational Geometry, 6(3):485-524, 1991.
Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994.
Bernard Chazelle, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, and Jack Snoeyink. Computing a face in an arrangement of line segments and related problems. SIAM Journal on Computing, 22(6):1286-1302, 1993.
Leonidas Guibas, John 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 and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989.
John Hershberger and Subhash Suri. Matrix searching with the shortest-path metric. SIAM Journal on Computing, 26(6):1612-1634, 1997.
Ferran Hurtado, Vera Sacristán, and Godfried Toussaint. Some constrained minimax and maximin location problems. Studies in Locational Analysis, 15:17-35, 2000.
David Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983.
D.T. Lee and V.B. Wu. Multiplicative weighted farthest neighbor Voronoi diagrams in the plane. In Proc. International Workshop on Discrete Mathematics and Algorithms, pages 154-168, 1993.
Nimrod Megiddo. Linear-time algorithms for linear programming in ℝ³ and related problems. SIAM Journal on Computing, 12(4):759-776, 1983.
Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The farthest-point geodesic Voronoi diagram of points on the boundary of a simple polygon. To appear in Proc. 32nd International Symposium on Computational Geometry (SoCG 2016), 2016.
Richard Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discrete &Computational Geometry, 4(6):611-626, 1989.
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Time-Space Trade-offs for Triangulating a Simple Polygon
An s-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses O(s) additional words of space. We give a randomized s-workspace algorithm for triangulating a simple polygon P of n vertices, for any s up to n. The algorithm runs in O(n^2/s+n(log s)log^5(n/s)) expected time using O(s) variables, for any s up to n. In particular, the algorithm runs in O(n^2/s) expected time for most values of s.
simple polygon
triangulation
shortest path
time-space trade-off
30:1-30:12
Regular Paper
Boris
Aronov
Boris Aronov
Matias
Korman
Matias Korman
Simon
Pratt
Simon Pratt
André
van Renssen
André van Renssen
Marcel
Roeloffzen
Marcel Roeloffzen
10.4230/LIPIcs.SWAT.2016.30
B. Aronov, M. Korman, S. Pratt, A. van Renssen, and M. Roeloffzen. Time-space trade-offs for triangulating a simple polygon. CoRR, abs/1509.07669, 2015. URL: http://arxiv.org/abs/1509.07669.
http://arxiv.org/abs/1509.07669
T. Asano, K. Buchin, M. Buchin, M. Korman, W. Mulzer, G. Rote, and A. Schulz. Memory-constrained algorithms for simple polygons. Computational Geometry: Theory and Applications, 46(8):959-969, 2013.
T. Asano and D. Kirkpatrick. Time-space tradeoffs for all-nearest-larger-neighbors problems. In Proc. 13th Int. Conf. Algorithms and Data Structures (WADS), pages 61-72, 2013.
T. Asano, W. Mulzer, G. Rote, and Y. Wang. Constant-work-space algorithms for geometric problems. Journal of Computational Geometry, 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. URL: http://dx.doi.org/10.1007/s00453-014-9893-5.
http://dx.doi.org/10.1007/s00453-014-9893-5
L. Barba, M. Korman, S. Langerman, and R. I. Silveira. Computing the visibility polygon using few variables. Computational Geometry: Theory and Applications, 47(9):918-926, 2013.
B. Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6:485-524, 1991. URL: http://dx.doi.org/10.1007/BF02574703.
http://dx.doi.org/10.1007/BF02574703
S. Har-Peled. Shortest path in a polygon using sublinear space. In Proceedings of the 31st International Symposium on Compututational Geometry (SoCG), pages 111-125, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.111.
http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.111
M. Korman. Memory-constrained algorithms. In Ming-Yang Kao, editor, Encyclopedia of Algorithms, pages 1-7. Springer Berlin Heidelberg, 2015. URL: http://dx.doi.org/10.1007/978-3-642-27848-8_586-1.
http://dx.doi.org/10.1007/978-3-642-27848-8_586-1
M. Korman, W. Mulzer, M. Roeloffzen, A. v. Renssen, P. Seiferth, and Y. Stein. Time-space trade-offs for triangulations and voronoi diagrams. In Proc. 14th Int. Conf. Algorithms and Data Structures (WADS), pages 482-494, 2015.
J. E. Savage. Models of Computation: Exploring the Power of Computing. Addison-Wesley, 1998.
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Excluded Grid Theorem: Improved and Simplified (Invited Talk)
One of the key results in Robertson and Seymour's seminal work on graph minors is the Excluded Grid Theorem. The theorem states that there is a function f, such that for every positive integer g, every graph whose treewidth is at least f(g) contains the (gxg)-grid as a minor. This theorem has found many applications in graph theory and algorithms. An important open question is establishing tight bounds on f(g) for which the theorem holds. Robertson and Seymour showed that f(g)>= \Omega(g^2 log g), and this remains the best current lower bound on f(g). Until recently, the best upper bound was super-exponential in g. In this talk, we will give an overview of a recent sequence of results, that has lead to the best current upper bound of f(g)=O(g^{19}polylog(g)). We will also survey some connections to algorithms for graph routing problems.
Graph Minor Theory
Excluded Grid Theorem
Graph Routing
31:1-31:1
Invited Talk
Julia
Chuzhoy
Julia Chuzhoy
10.4230/LIPIcs.SWAT.2016.31
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The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems (Invited Talk)
Given a directed graph G and a list (s_1,t_1), ..., (s_k,t_k) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s_i-> t_i path for every 1<= i <= k. Feldman and Ruhl presented an n^{O(k)} time algorithm for the problem, which shows that it is polynomial-time solvable for every fixed number k of demands. There are special cases of the problem that can be solved much more efficiently: for example, the special case Directed Steiner Tree (when we ask for paths from a root r to terminals t_1, ..., t_k) is known to be fixed-parameter tractable parameterized by the number of terminals, that is, algorithms with running time of the form f(k)*n^{O(1)} exist for the problem. On the other hand, the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t_i to every other t_j) is known to be W[1]-hard parameterized by the number of terminals, hence it is unlikely to be fixed-parameter tractable. In the talk, we survey results on parameterized algorithms for special cases of Directed Steiner Network, including a recent complete classification result (joint work with Andreas Feldmann) that systematically explores the complexity landscape of directed Steiner problems to fully understand which special cases are FPT or W[1]-hard.
Directed Steiner Tree
Directed Steiner Network
fixed-parameter tractability
dichotomy
32:1-32:1
Invited Talk
Dániel
Marx
Dániel Marx
10.4230/LIPIcs.SWAT.2016.32
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Computation as a Scientific Weltanschauung (Invited Talk)
Computation as a mechanical reality is young - almost exactly seventy years of age - and yet the spirit of computation can be traced several millennia back. Any moderately advanced civilization depends on calculation (for inventory, taxation, navigation, land partition, among many others) - our civilization is the first one that is conscious of this reliance.
Computation has also been central to science for centuries. This is most immediately apparent in the case of mathematics: the idea of the algorithm as a mathematical object of some significance was pioneered by Euclid in the 4th century BC, and advanced by Archimedes a century later. But computation plays an important role in virtually all sciences: natural, life, or social. Implicit algorithmic processes are present in the great objects of scientific inquiry - the cell, the universe, the market, the brain - as well as in the models developed by scientists over the centuries for studying them. This brings about a very recent - merely a few decades old - mode of scientific inquiry, which is sometime referred to as the lens of computation: When students of computation revisit central problems in science from the computational viewpoint, often unexpected progress results. This has happened in statistical physics through the study of phase transitions in terms of the convergence of Markov chain-Monte Carlo algorithms, and in quantum mechanics through quantum computing.
This talk will focus on three other manifestations of this phenomenon. Almost a decade ago, ideas and methodologies from computational complexity revealed a subtle conceptual flaw in the solution concept of Nash equilibrium, which lies at the foundations of modern economic thought. In the study of evolution, a new understanding of century-old questions has been achieved through surprisingly algorithmic ideas. Finally, current work in theoretical neuroscience suggests that the algorithmic point of view may be invaluable in the central scientific question of our era, namely understanding how behavior and cognition emerge from the structure and activity of neurons and synapses.
Lens of computation
Nash equilibrium
neuroscience
33:1-33:1
Invited Talk
Christos H.
Papadimitriou
Christos H. Papadimitriou
10.4230/LIPIcs.SWAT.2016.33
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Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network.
We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves.
Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices.
Network Augmentation
Shortcuts
Diameter
Paths
Cycles
27:1-27:14
Regular Paper
Jean-Lou
De Carufel
Jean-Lou De Carufel
Carsten
Grimm
Carsten Grimm
Anil
Maheshwari
Anil Maheshwari
Michiel
Smid
Michiel Smid
10.4230/LIPIcs.SWAT.2016.27
Jean-Lou De Carufel, Carsten Grimm, Anil Maheshwari, and Michiel Smid. Minimizing the continuous diameter when augmenting paths and cycles with shortcuts. This is the full version of this paper., 2015. URL: http://arxiv.org/abs/1512.02257.
http://arxiv.org/abs/1512.02257
Victor Chepoi and Yann Vaxès. Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica, 33(2):243-262, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0113-8.
http://dx.doi.org/10.1007/s00453-001-0113-8
Mohammad Farshi, Panos Giannopoulos, and Joachim Gudmundsson. Improving the stretch factor of a geometric network by edge augmentation. SIAM Journal on Computing, 38(1):226-240, 2008. URL: http://dx.doi.org/10.1137/050635675.
http://dx.doi.org/10.1137/050635675
Fabrizio Frati, Serge Gaspers, Joachim Gudmundsson, and Luke Mathieson. Augmenting graphs to minimize the diameter. Algorithmica, 72(4):995-1010, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9886-4.
http://dx.doi.org/10.1007/s00453-014-9886-4
Yong Gao, Donovan R. Hare, and James Nastos. The parametric complexity of graph diameter augmentation. Discrete Applied Mathematics, 161(10-11):1626-1631, 2013. URL: http://dx.doi.org/10.1016/j.dam.2013.01.016.
http://dx.doi.org/10.1016/j.dam.2013.01.016
Ulrike Große, Joachim Gudmundsson, Christian Knauer, Michiel Smid, and Fabian Stehn. Fast algorithms for diameter-optimally augmenting paths. In 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015), pages 678-688, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_55.
http://dx.doi.org/10.1007/978-3-662-47672-7_55
Chung-Lun Li, S. Thomas McCormick, and David Simchi-Levi. On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems. Operations Research Letters, 11(5):303-308, 1992. URL: http://dx.doi.org/10.1016/0167-6377(92)90007-P.
http://dx.doi.org/10.1016/0167-6377(92)90007-P
Jun Luo and Christian Wulff-Nilsen. Computing best and worst shortcuts of graphs embedded in metric spaces. In 19th International Symposium on Algorithms and Computation (ISAAC 2008), pages 764-775, 2008. URL: http://dx.doi.org/10.1007/978-3-540-92182-0_67.
http://dx.doi.org/10.1007/978-3-540-92182-0_67
Anneke A. Schoone, Hans L. Bodlaender, and Jan van Leeuwen. Diameter increase caused by edge deletion. Journal of Graph Theory, 11(3):409-427, 1987. URL: http://dx.doi.org/10.1002/jgt.3190110315.
http://dx.doi.org/10.1002/jgt.3190110315
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