2nd Symposium on Simplicity in Algorithms (SOSA 2019), SOSA 2019, January 8-9, 2019, San Diego, CA, USA
SOSA 2019
January 8-9, 2019
San Diego, CA, USA
Open Access Series in Informatics
OASIcs
https://www.dagstuhl.de/dagpub/2190-6807
https://dblp.org/db/series/oasics
2190-6807
Jeremy T.
Fineman
Jeremy T. Fineman
Michael
Mitzenmacher
Michael Mitzenmacher
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
69
2019
978-3-95977-099-6
https://www.dagstuhl.de/dagpub/978-3-95977-099-6
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:x
Front Matter
Jeremy T.
Fineman
Jeremy T. Fineman
Michael
Mitzenmacher
Michael Mitzenmacher
10.4230/OASIcs.SOSA.2019.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Isotonic Regression by Dynamic Programming
For a given sequence of numbers, we want to find a monotonically increasing sequence of the same length that best approximates it in the sense of minimizing the weighted sum of absolute values of the differences. A conceptually easy dynamic programming approach leads to an algorithm with running time O(n log n). While other algorithms with the same running time are known, our algorithm is very simple. The only auxiliary data structure that it requires is a priority queue. The approach extends to other error measures.
Convex functions
dynamic programming
convex hull
isotonic regression
1:1-1:18
Regular Paper
Günter
Rote
Günter Rote
10.4230/OASIcs.SOSA.2019.1
R. K. Ahuja and J. B. Orlin. A fast scaling algorithm for minimizing separable convex functions subject to chain constraints. Operations Research, 49:784-789, 2001. URL: http://dx.doi.org/10.1287/opre.49.5.784.10601.
http://dx.doi.org/10.1287/opre.49.5.784.10601
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Heather Booth and Robert Endre Tarjan. Finding the minimum-cost maximum flow in a series-parallel network. J. Algorithms, 15:416-446, 1993. URL: http://dx.doi.org/10.1006/jagm.1993.1048.
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Creative Commons Attribution 3.0 Unported license
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An Illuminating Algorithm for the Light Bulb Problem
The Light Bulb Problem is one of the most basic problems in data analysis. One is given as input n vectors in {-1,1}^d, which are all independently and uniformly random, except for a planted pair of vectors with inner product at least rho * d for some constant rho > 0. The task is to find the planted pair. The most straightforward algorithm leads to a runtime of Omega(n^2). Algorithms based on techniques like Locality-Sensitive Hashing achieve runtimes of n^{2 - O(rho)}; as rho gets small, these approach quadratic.
Building on prior work, we give a new algorithm for this problem which runs in time O(n^{1.582} + nd), regardless of how small rho is. This matches the best known runtime due to Karppa et al. Our algorithm combines techniques from previous work on the Light Bulb Problem with the so-called `polynomial method in algorithm design,' and has a simpler analysis than previous work. Our algorithm is also easily derandomized, leading to a deterministic algorithm for the Light Bulb Problem with the same runtime of O(n^{1.582} + nd), improving previous results.
Light Bulb Problem
Polynomial Method
Finding Correlations
2:1-2:11
Regular Paper
Josh
Alman
Josh Alman
10.4230/OASIcs.SOSA.2019.2
Josh Alman, Timothy M Chan, and Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In FOCS, 2016.
Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In FOCS, 2015.
Moses S Charikar. Similarity estimation techniques from rounding algorithms. In STOC, 2002.
Lijie Chen. On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product. In CCC, 2018.
Moshe Dubiner. Bucketing coding and information theory for the statistical high-dimensional nearest-neighbor problem. IEEE Transactions on Information Theory, 56(8):4166-4179, 2010.
Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. On agnostic learning of parities, monomials, and halfspaces. SIAM Journal on Computing, 39(2):606-645, 2009.
Oded Goldreich and Avi Wigderson. Derandomization that is rarely wrong from short advice that is typically good. Lecture notes in computer science, 2483:209-223, 2002.
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Matti Karppa, Petteri Kaski, and Jukka Kohonen. A faster subquadratic algorithm for finding outlier correlations. In SODA, 2016.
Matti Karppa, Petteri Kaski, Jukka Kohonen, and Padraig Ó Catháin. Explicit Correlation Amplifiers for Finding Outlier Correlations in Deterministic Subquadratic Time. In ESA, 2016.
François Le Gall. Powers of tensors and fast matrix multiplication. In ISSAC, 2014.
Shachar Lovett. Computing Polynomials with Few Multiplications. Theory of Computing, 7(1):185-188, 2011.
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Ramamohan Paturi, Sanguthevar Rajasekaran, and John Reif. The light bulb problem. Information and Computation, 117(2):187-192, 1995.
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Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13(4):354-356, 1969.
Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and the closest pair problem. Journal of the ACM, 62(2):13, 2015.
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R Ryan Williams. Natural proofs versus derandomization. SIAM Journal on Computing, 45(2):497-529, 2016.
Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In STOC, 2012.
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Simple Concurrent Labeling Algorithms for Connected Components
We present new concurrent labeling algorithms for finding connected components, and we study their theoretical efficiency. Even though many such algorithms have been proposed and many experiments with them have been done, our algorithms are simpler. We obtain an O(lg n) step bound for two of our algorithms using a novel multi-round analysis. We conjecture that our other algorithms also take O(lg n) steps but are only able to prove an O(lg^2 n) bound. We also point out some gaps in previous analyses of similar algorithms. Our results show that even a basic problem like connected components still has secrets to reveal.
Connected Components
Concurrent Algorithms
3:1-3:20
Regular Paper
Sixue
Liu
Sixue Liu
Robert E.
Tarjan
Robert E. Tarjan
10.4230/OASIcs.SOSA.2019.3
Alexandr Andoni, Zhao Song, Clifford Stein, Zhengyu Wang, and Peilin Zhong. Parallel Graph Connectivity in Log Diameter Rounds. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 674-685, 2018.
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Paul Beame, Paraschos Koutris, and Dan Suciu. Communication Steps for Parallel Query Processing. J. ACM, 64(6):40:1-40:58, 2017.
Paul Burkhardt. Graph connectivity in log-diameter steps using label propagation. CoRR, 2018. URL: http://arxiv.org/abs/1808.06705.
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John Greiner. A Comparison of Parallel Algorithms for Connected Components. In SPAA, pages 16-25, 1994.
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A Framework for Searching in Graphs in the Presence of Errors
We consider a problem of searching for an unknown target vertex t in a (possibly edge-weighted) graph. Each vertex-query points to a vertex v and the response either admits that v is the target or provides any neighbor s of v that lies on a shortest path from v to t. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability p<1/2 and a correct one with probability 1-p).
We study this problem both with adversarial errors and independent noise models. First, we show an algorithm that needs at most (log_2 n)/(1 - H(r)) queries in case of adversarial errors, where the adversary is bounded with its rate of errors by a known constant r<1/2. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking an amortization argument. We then show that our algorithm coupled with a Chernoff bound argument leads to a simpler algorithm for the independent noise model and has a query complexity that is both simpler and asymptotically better than the one of Emamjomeh-Zadeh et al. [STOC 2016].
Our approach has a wide range of applications. First, it improves and simplifies the Robust Interactive Learning framework proposed by Emamjomeh-Zadeh and Kempe [NIPS 2017]. Secondly, performing analogous analysis for edge-queries (where a query to an edge e returns its endpoint that is closer to the target) we actually recover (as a special case) a noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon an algorithm for searching of unbounded domains due to Aslam and Dhagat [STOC 1991].
graph algorithms
noisy binary search
query complexity
reliability
4:1-4:17
Regular Paper
Dariusz
Dereniowski
Dariusz Dereniowski
Stefan
Tiegel
Stefan Tiegel
Przemyslaw
Uznanski
Przemyslaw Uznanski
Daniel
Wolleb-Graf
Daniel Wolleb-Graf
10.4230/OASIcs.SOSA.2019.4
Dana Angluin. Queries and Concept Learning. Machine Learning, 2(4):319-342, 1987. URL: http://dx.doi.org/10.1007/BF00116828.
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Argyrios Deligkas, George B. Mertzios, and Paul G. Spirakis. Binary Search in Graphs Revisited. In MFCS, pages 20:1-20:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2017.20.
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Dariusz Dereniowski. Edge ranking and searching in partial orders. Discrete Applied Mathematics, 156(13):2493-2500, 2008. URL: http://dx.doi.org/10.1016/j.dam.2008.03.007.
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Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznański, and Mengchuan Zou. Approximation Strategies for Generalized Binary Search in Weighted Trees. In ICALP, pages 84:1-84:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.84.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.84
Dariusz Dereniowski and Adam Nadolski. Vertex rankings of chordal graphs and weighted trees. Inf. Process. Lett., 98(3):96-100, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2005.12.006.
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Ehsan Emamjomeh-Zadeh and David Kempe. A General Framework for Robust Interactive Learning. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pages 7085-7094, 2017. URL: http://papers.nips.cc/paper/7283-a-general-framework-for-robust-interactive-learning.
http://papers.nips.cc/paper/7283-a-general-framework-for-robust-interactive-learning
Ehsan Emamjomeh-Zadeh, David Kempe, and Vikrant Singhal. Deterministic and probabilistic binary search in graphs. In STOC, pages 519-532, 2016. URL: http://dx.doi.org/10.1145/2897518.2897656.
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Nicolas Hanusse, Dimitris J. Kavvadias, Evangelos Kranakis, and Danny Krizanc. Memoryless search algorithms in a network with faulty advice. Theor. Comput. Sci., 402(2-3):190-198, 2008. URL: http://dx.doi.org/10.1016/j.tcs.2008.04.034.
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Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Selection from Heaps, Row-Sorted Matrices, and X+Y Using Soft Heaps
We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X+Y, where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal "output-sensitive" algorithm that performs only O(m + sum_{i=1}^m log(k_i+1)) comparisons, where k_i is the number of items of the i-th list that belong to the overall set of k smallest items.
selection
soft heap
5:1-5:21
Regular Paper
Haim
Kaplan
Haim Kaplan
László
Kozma
László Kozma
Or
Zamir
Or Zamir
Uri
Zwick
Uri Zwick
10.4230/OASIcs.SOSA.2019.5
Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time Bounds for Selection. J. Comput. Syst. Sci., 7(4):448-461, 1973. URL: http://dx.doi.org/10.1016/S0022-0000(73)80033-9.
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Bernard Chazelle. The soft heap: an approximate priority queue with optimal error rate. J. ACM, 47(6):1012-1027, 2000. URL: http://dx.doi.org/10.1145/355541.355554.
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Dorit Dor and Uri Zwick. Selecting the Median. SIAM Journal on Computing, 28(5):1722-1758, 1999. URL: http://dx.doi.org/10.1137/S0097539795288611.
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Dorit Dor and Uri Zwick. Median Selection Requires (2+ε)n Comparisons. SIAM Journal on Discrete Mathematics, 14(3):312-325, 2001. URL: http://dx.doi.org/10.1137/S0895480199353895.
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David Eppstein. Finding the k Shortest Paths. SIAM J. Comput., 28(2):652-673, 1998. URL: http://dx.doi.org/10.1137/S0097539795290477.
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Donald B. Johnson and Samuel D. Kashdan. Lower Bounds for Selection in X+Y and Other Multisets. J. ACM, 25(4):556-570, 1978. URL: http://dx.doi.org/10.1145/322092.322097.
http://dx.doi.org/10.1145/322092.322097
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Daniel M. Kane, Shachar Lovett, and Shay Moran. Near-optimal linear decision trees for k-SUM and related problems. In Proc. of 50th STOC, pages 554-563, 2018. URL: http://dx.doi.org/10.1145/3188745.3188770.
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Haim Kaplan, Robert Endre Tarjan, and Uri Zwick. Soft Heaps Simplified. SIAM Journal on Computing, 42(4):1660-1673, 2013. URL: http://dx.doi.org/10.1137/120880185.
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Approximating Optimal Transport With Linear Programs
In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in faster additive approximation algorithms for optimal transport.
optimal transport
fast approximations
linear programming
6:1-6:9
Regular Paper
Kent
Quanrud
Kent Quanrud
10.4230/OASIcs.SOSA.2019.6
Zeyuan Allen Zhu and Lorenzo Orecchia. Nearly-Linear Time Positive LP Solver with Faster Convergence Rate. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 229-236, 2015.
Jason Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pages 1961-1971, 2017.
Jose Blanchet, Arun Jambulapati, Carson Kent, and Aaron Sidford. Towards Optimal Running Times for Optimal Transport. CoRR, abs/1810.07717, 2018. URL: http://arxiv.org/abs/1810.07717.
http://arxiv.org/abs/1810.07717
Deeparnab Chakrabarty and Sanjeev Khanna. Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling. In 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, New Orleans, LA, USA, pages 4:1-4:11, 2018.
Chandra Chekuri and Kent Quanrud. Near-Linear Time Approximation Schemes for some Implicit Fractional Packing Problems. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 801-820, 2017.
Chandra Chekuri and Kent Quanrud. Randomized MWU for Positive LPs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 358-377, 2018.
Michael B. Cohen, Aleksander Madry, Dimitris Tsipras, and Adrian Vladu. Matrix Scaling and Balancing via Box Constrained Newton’s Method and Interior Point Methods. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 902-913, 2017.
Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States., pages 2292-2300, 2013.
Pavel Dvurechensky, Alexander Gasnikov, and Alexey Kroshnin. Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn’s Algorithm. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, pages 1366-1375, 2018.
Christos Koufogiannakis and Neal E. Young. A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs. Algorithmica, 70(4):648-674, 2014. Preliminary version in FOCS 2007.
Yin Tat Lee and Aaron Sidford. Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(√rank) Iterations and Faster Algorithms for Maximum Flow. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 424-433, 2014.
Michael W. Mahoney, Satish Rao, Di Wang, and Peng Zhang. Approximating the Solution to Mixed Packing and Covering LPs in Parallel Õ(ε^-3) time. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 52:1-52:14, 2016.
Jonah Sherman. Generalized Preconditioning and Undirected Minimum-Cost Flow. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 772-780, 2017.
Richard Sinkhorn and Paul Knopp. Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics, 21(2):343-348, 1967.
Cédric Villani. Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, 2003.
Cédric Villani. Optimal transport: old and new, volume 338 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 2009.
Neal E. Young. Nearly Linear-Time Approximation Schemes for Mixed Packing/Covering and Facility-Location Linear Programs. CoRR, abs/1407.3015, 2014. URL: http://arxiv.org/abs/1407.3015.
http://arxiv.org/abs/1407.3015
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LP Relaxation and Tree Packing for Minimum k-cuts
Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008].
In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008].
k-cut
LP relaxation
tree packing
7:1-7:18
Regular Paper
Chandra
Chekuri
Chandra Chekuri
Kent
Quanrud
Kent Quanrud
Chao
Xu
Chao Xu
10.4230/OASIcs.SOSA.2019.7
Ajit Agrawal, Philip Klein, and R Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 24(3):440-456, 1995.
Francisco Barahona. On the k-cut problem. Operations Research Letters, 26(3):99-105, 2000.
Chandra Chekuri, Sudipto Guha, and Joseph Naor. The Steiner k-cut problem. SIAM Journal on Discrete Mathematics, 20(1):261-271, 2006.
Chandra Chekuri and Kent Quanrud. Near-linear time approximation schemes for some implicit fractional packing problems. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 801-820. SIAM, 2017.
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Rodney G. Downey, Vladimir Estivill-Castro, Michael R. Fellows, Elena Prieto-Rodriguez, and Frances A. Rosamond. Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. Electr. Notes Theor. Comput. Sci., 78:209-222, 2003.
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O. Goldschmidt and D.S. Hochbaum. A polynomial algorithm for the k-cut problem for fixed k. Mathematics of Operations Research, pages 24-37, 1994.
Anupam Gupta, Euiwoong Lee, and Jason Li. Faster Exact and Approximate Algorithms for k-Cut. In Proceedings of IEEE FOCS, 2018.
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David R. Karger. Minimum Cuts in Near-linear Time. J. ACM, 47(1):46-76, January 2000. URL: http://dx.doi.org/10.1145/331605.331608.
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David R Karger and Clifford Stein. A new approach to the minimum cut problem. Journal of the ACM (JACM), 43(4):601-640, 1996.
Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 665-674. ACM, 2015.
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J Naor and Yuval Rabani. Tree Packing and Approximating k-Cuts. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, volume 103, page 26. SIAM, 2001.
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Kent Quanrud. Fast and Deterministic Approximations for k-Cut. CoRR, abs/1807.07143, 2018. URL: http://arxiv.org/abs/1807.07143.
http://arxiv.org/abs/1807.07143
R Ravi and Amitabh Sinha. Approximating k-cuts using network strength as a lagrangean relaxation. European Journal of Operational Research, 186(1):77-90, 2008.
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Mikkel Thorup. Fully-dynamic min-cut. Combinatorica, 27(1):91-127, 2007.
Mikkel Thorup. Minimum k-way cuts via deterministic greedy tree packing. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages 159-166. ACM, 2008.
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On Primal-Dual Circle Representations
The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a contact representation by circles. The theorem has been generalized in various ways. The most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. We present a simple and elegant elementary proof of this result.
Disk packing
planar graphs
contact representation
8:1-8:18
Regular Paper
Stefan
Felsner
Stefan Felsner
Günter
Rote
Günter Rote
10.4230/OASIcs.SOSA.2019.8
Nieke Aerts and Stefan Felsner. Straight Line Triangle Representations. Discr. and Comput. Geom., 57:257-280, 2017. URL: http://dx.doi.org/10.1007/s00454-016-9850-y.
http://dx.doi.org/10.1007/s00454-016-9850-y
E. M. Andreev. Convex polyhedra in Lobačevskiĭspaces. Mat. Sb. (N.S.), 81 (123):445-478, 1970. English: Math. USSR, Sb. 10, 413-440 (1971).
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Marshall W. Bern and David Eppstein. Quadrilateral Meshing by Circle Packing. Int. J. Comput. Geometry Appl., 10:347-360, 2000.
Alexander I. Bobenko and Boris A. Springborn. Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc., 356:659-689, 2004.
G. R. Brightwell and E. R. Scheinerman. Representations of Planar Graphs. SIAM J. Discr. Math., 6(2):214-229, 1993.
Yves Colin de Verdière. Empilements de cercles: convergence d'une méthode de point fixe. Forum Math., 1:395-402, 1989.
Yves Colin de Verdière. Un principe variationnel pour les empilements de cercles. Invent. Math., 104:655-669, 1991.
Hubert de Fraysseix, Patrice Ossona de Mendez, and Pierre Rosenstiehl. On Triangle Contact Graphs. Comb., Probab. and Comput., 3(02):233-246, 1994.
Olivier Devillers, Giuseppe Liotta, Franco P. Preparata, and Roberto Tamassia. Checking the convexity of polytopes and the planarity of subdivisions. Comput. Geom., 11(3-4):187-208, 1998. URL: http://dx.doi.org/10.1016/S0925-7721(98)00039-X.
http://dx.doi.org/10.1016/S0925-7721(98)00039-X
David Eppstein. Diamond-Kite Meshes: Adaptive Quadrilateral Meshing and Orthogonal Circle Packing. In Proc. Mesh. Roundtable, pages 261-277. Springer, 2012.
David Eppstein. A Möbius-invariant power diagram and its applications to soap bubbles and planar Lombardi drawing. Discrete Comput. Geom., 52:515-550, 2014.
Stefan Felsner. Rectangle and Square Representations of Planar Graphs. In J. Pach, editor, Thirty Essays in Geometric Graph Theory, pages 213-248. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_12.
http://dx.doi.org/10.1007/978-1-4614-0110-0_12
Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, and Manfred Scheucher. Strongly Monotone Drawings of Planar Graphs. In Proc. 32nd Intern. Symp. Comput. Geom. (SoCG 2016), volume 51 of LIPIcs, pages 37:1-15, 2016.
Stefan Felsner, Hendrik Schrezenmaier, and Raphael Steiner. Equiangular polygon contact representations, 2017. http://page.math.tu-berlin.de/~felsner/Paper/kgons.pdf.
Stefan Felsner, Hendrik Schrezenmaier, and Raphael Steiner. Pentagon Contact Representations. Electronic J. Combin., 25(3):article #P3.39, 38 pp., 2018.
Daniel Gonçalves, Benjamin Lévêque, and Alexandre Pinlou. Triangle Contact Representations and Duality. Discr. and Comput. Geom., 48(1):239-254, 2012.
S. Har-Peled. A Simple Proof of the Existence of a Planar Separator. arXiv:http://arxiv.org/abs/1105.0103, 2011.
http://arxiv.org/abs/1105.0103
Nora Hartsfield and Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introduction. Academic Press, 1990.
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Paul Koebe. Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akad. Leipzig, Math.-Phys. Klasse, 88:141-164, 1936.
László Lovász. Geometric Representations of Graphs. http://web.cs.elte.hu/~lovasz/geomrep.pdf, December 2009. Draft version.
Gary L. Miller, Shang-Hua Teng, William Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, 1997.
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Bojan Mohar. Circle packings of maps - the Euclidean case. Rend. Sem. Mat. Fis. Milano, 67:191-206, 2000.
David Orden, Günter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, and Walter Whiteley. Non-crossing frameworks with non-crossing reciprocals. Discrete and Computational Geometry, 32:567-600, 2004. URL: http://dx.doi.org/10.1007/s00454-004-1139-x.
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Oded Schramm. Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps. arXiv:http://arxiv.org/abs/0709.0710, 2007. Modified version of PhD thesis from 1990.
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Kenneth Stephenson. Circle packing: a mathematical tale. Notices Amer. Math. Soc., 50:1376-1388, 2003.
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Asymmetric Convex Intersection Testing
We consider asymmetric convex intersection testing (ACIT).
Let P subset R^d be a set of n points and H a set of n halfspaces in d dimensions. We denote by {ch(P)} the polytope obtained by taking the convex hull of P, and by {fh(H)} the polytope obtained by taking the intersection of the halfspaces in H. Our goal is to decide whether the intersection of H and the convex hull of P are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before.
We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on n and m, whose running time depends reasonably on the dimension d.
polytope intersection
LP-type problem
randomized algorithm
9:1-9:14
Regular Paper
Luis
Barba
Luis Barba
Wolfgang
Mulzer
Wolfgang Mulzer
10.4230/OASIcs.SOSA.2019.9
Luis Barba and Stefan Langerman. Optimal detection of intersections between convex polyhedra. In Proc. 26th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 1641-1654, 2015.
Timothy M. Chan. An optimal randomized algorithm for maximum Tukey depth. In Proc. 15th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 430-436, 2004.
Timothy M Chan. Improved deterministic algorithms for linear programming in low dimensions. ACM Trans. Algorithms, 14(3):30, 2018.
Timothy M Chan. Personal communication, 2018.
B. Chazelle and D. Dobkin. Intersection of Convex Objects in Two and Three Dimensions. J. ACM, 34(1):1-27, January 1987.
Bernard Chazelle. An optimal algorithm for intersecting three-dimensional convex polyhedra. SIAM J. Comput., 21:586-591, 1992.
Bernard Chazelle. An Optimal Convex Hull Algorithm in Any Fixed Dimension. Discrete Comput. Geom., 10:377-409, 1993.
Bernard Chazelle. The Discrepancy Method - Randomness and Complexity. Cambridge University Press, 2001.
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Vašek Chvátal. Linear programming. A Series of Books in the Mathematical Sciences. W. H. Freeman, 1983.
Kenneth L. Clarkson. A Las Vegas Algorithm for Linear Programming When the Dimension Is Small. In Proc. 29th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 452-456, 1988.
Kenneth L. Clarkson and Peter W. Shor. Application of Random Sampling in Computational Geometry, II. Discrete Comput. Geom., 4:387-421, 1989.
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Sariel Har-Peled. Geometric approximation algorithms. American Mathematical Society, 2011.
Sanjiv Kapoor and Pravin M. Vaidya. Fast Algorithms for Convex Quadratic Programming and Multicommodity Flows. In Proc. 18th Annu. ACM Sympos. Theory Comput. (STOC), pages 147-159, 1986.
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Micha Sharir and Emo Welzl. A combinatorial bound for linear programming and related problems. Proc. 9th Sympos. Theoret. Aspects Comput. Sci. (STACS), pages 567-579, 1992.
Günter M. Ziegler. Lectures on Polytopes. Springer-Verlag, 1995.
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Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems
We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space (X,d) (possibly arising from a graph) and a subset of terminals K subset X, and the goal is to partition the points X such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric 0-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric 0-Extension [Englert et al., SICOMP 2014].
A natural approach is to cluster each point with its closest terminal, which would partition X into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.
Clustering
Steiner point removal
Zero extension
Doubling dimension
Relaxed voronoi
10:1-10:14
Regular Paper
Arnold
Filtser
Arnold Filtser
Robert
Krauthgamer
Robert Krauthgamer
Ohad
Trabelsi
Ohad Trabelsi
10.4230/OASIcs.SOSA.2019.10
Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '04, pages 1079-1087, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982952.
http://dl.acm.org/citation.cfm?id=982792.982952
A. Basu and A. Gupta. Steiner Point Removal in Graph Metrics. Unpublished Manuscript, available from http://www.math.ucdavis.edu/~abasu/papers/SPR.pdf, 2008.
http://www.math.ucdavis.edu/~abasu/papers/SPR.pdf
Gruia Călinescu, Howard J. Karloff, and Yuval Rabani. Approximation Algorithms for the 0-Extension Problem. SIAM J. Comput., 34(2):358-372, 2004. URL: http://dx.doi.org/10.1137/S0097539701395978.
http://dx.doi.org/10.1137/S0097539701395978
T.-H. Chan, Donglin Xia, Goran Konjevod, and Andrea Richa. A Tight Lower Bound for the Steiner Point Removal Problem on Trees. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th International Conference on Randomization and Computation, APPROX'06/RANDOM'06, pages 70-81, 2006. URL: http://dx.doi.org/10.1007/11830924_9.
http://dx.doi.org/10.1007/11830924_9
Yun Kuen Cheung. Steiner Point Removal - Distant Terminals Don't (Really) Bother. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 1353-1360, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.89.
http://dx.doi.org/10.1137/1.9781611975031.89
Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The Complexity of Multiway Cuts (Extended Abstract). In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992, pages 241-251, 1992. URL: http://dx.doi.org/10.1145/129712.129736.
http://dx.doi.org/10.1145/129712.129736
Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Räcke, Inbal Talgam-Cohen, and Kunal Talwar. Vertex Sparsifiers: New Results from Old Techniques. SIAM J. Comput., 43(4):1239-1262, 2014. URL: http://dx.doi.org/10.1137/130908440.
http://dx.doi.org/10.1137/130908440
Jittat Fakcharoenphol, Chris Harrelson, Satish Rao, and Kunal Talwar. An improved approximation algorithm for the 0-extension problem. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '03, pages 257-265, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644153.
http://dl.acm.org/citation.cfm?id=644108.644153
Arnold Filtser. Steiner Point Removal with Distortion O(log k). In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 1361-1373, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.90.
http://dx.doi.org/10.1137/1.9781611975031.90
Arnold Filtser. Steiner Point Removal with distortion O(log k), using the Noisy-Voronoi algorithm. CoRR, abs/1808.02800, 2018. URL: http://arxiv.org/abs/1808.02800.
http://arxiv.org/abs/1808.02800
Michael L. Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34(3):596-615, 1987. URL: http://dx.doi.org/10.1145/28869.28874.
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Teofilo F. Gonzalez. Clustering to Minimize the Maximum Intercluster Distance. Theor. Comput. Sci., 38:293-306, 1985. URL: http://dx.doi.org/10.1016/0304-3975(85)90224-5.
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Anupam Gupta. Steiner Points in Tree Metrics Don'T (Really) Help. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '01, pages 220-227, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365448.
http://dl.acm.org/citation.cfm?id=365411.365448
Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded Geometries, Fractals, and Low-Distortion Embeddings. In 44th Symposium on Foundations of Computer Science (FOCS 2003), pages 534-543, 2003. URL: http://dx.doi.org/10.1109/SFCS.2003.1238226.
http://dx.doi.org/10.1109/SFCS.2003.1238226
Anupam Gupta and Kunal Talwar. Random Rates for 0-Extension and Low-Diameter Decompositions. CoRR, abs/1307.5582, 2013. URL: http://arxiv.org/abs/1307.5582.
http://arxiv.org/abs/1307.5582
Lior Kamma, Robert Krauthgamer, and Huy L. Nguyen. Cutting Corners Cheaply, or How to Remove Steiner Points. SIAM J. Comput., 44(4):975-995, 2015. URL: http://dx.doi.org/10.1137/140951382.
http://dx.doi.org/10.1137/140951382
Alexander V. Karzanov. Minimum 0-Extensions of Graph Metrics. Eur. J. Comb., 19(1):71-101, 1998. URL: http://dx.doi.org/10.1006/eujc.1997.0154.
http://dx.doi.org/10.1006/eujc.1997.0154
Jon M. Kleinberg and Éva Tardos. Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields. J. ACM, 49(5):616-639, 2002. URL: http://dx.doi.org/10.1145/585265.585268.
http://dx.doi.org/10.1145/585265.585268
James R. Lee and Assaf Naor. Metric decomposition, smooth measures, and clustering. Unpublished Manuscript, available from https://www.math.nyu.edu/~naor/homepage%20files/cluster.pdf, 2003.
https://www.math.nyu.edu/~naor/homepage%20files/cluster.pdf
Gary L. Miller, Richard Peng, and Shen Chen Xu. Parallel graph decompositions using random shifts. In 25th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '13, pages 196-203, 2013. URL: http://dx.doi.org/10.1145/2486159.2486180.
http://dx.doi.org/10.1145/2486159.2486180
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Towards a Unified Theory of Sparsification for Matching Problems
In this paper, we present a construction of a "matching sparsifier", that is, a sparse subgraph of the given graph that preserves large matchings approximately and is robust to modifications of the graph. We use this matching sparsifier to obtain several new algorithmic results for the maximum matching problem:
- An almost (3/2)-approximation one-way communication protocol for the maximum matching problem, significantly simplifying the (3/2)-approximation protocol of Goel, Kapralov, and Khanna (SODA 2012) and extending it from bipartite graphs to general graphs.
- An almost (3/2)-approximation algorithm for the stochastic matching problem, improving upon and significantly simplifying the previous 1.999-approximation algorithm of Assadi, Khanna, and Li (EC 2017).
- An almost (3/2)-approximation algorithm for the fault-tolerant matching problem, which, to our knowledge, is the first non-trivial algorithm for this problem.
Our matching sparsifier is obtained by proving new properties of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (ICALP 2015; SODA 2016) - designed in the context of maintaining matchings in dynamic graphs - that identifies EDCS as an excellent choice for a matching sparsifier. This leads to surprisingly simple and non-technical proofs of the above results in a unified way. Along the way, we also provide a much simpler proof of the fact that an EDCS is guaranteed to contain a large matching, which may be of independent interest.
Maximum matching
matching sparsifiers
one-way communication complexity
stochastic matching
fault-tolerant matching
11:1-11:20
Regular Paper
Sepehr
Assadi
Sepehr Assadi
Aaron
Bernstein
Aaron Bernstein
10.4230/OASIcs.SOSA.2019.11
Noga Alon and Joel H Spencer. The probabilistic method. John Wiley &Sons, 2004.
Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab S. Mirrokni, and Cliff Stein. Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs. CoRR, abs/1711.03076. To appear in SODA 2019, 2017.
Sepehr Assadi, Sanjeev Khanna, and Yang Li. The Stochastic Matching Problem with (Very) Few Queries. In Proceedings of the 2016 ACM Conference on Economics and Computation, EC '16, Maastricht, The Netherlands, July 24-28, 2016, pages 43-60, 2016.
Sepehr Assadi, Sanjeev Khanna, and Yang Li. On Estimating Maximum Matching Size in Graph Streams. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1723-1742, 2017.
Sepehr Assadi, Sanjeev Khanna, and Yang Li. The Stochastic Matching Problem: Beating Half with a Non-Adaptive Algorithm. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017, pages 99-116, 2017.
Baruch Awerbuch. Complexity of Network Synchronization. J. ACM, 32(4):804-823, 1985.
Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault tolerant subgraph for single source reachability: generic and optimal. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 509-518, 2016.
Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 255-262, 2009.
Soheil Behnezhad, Alireza Farhadi, MohammadTaghi Hajiaghayi, and Nima Reyhani. Stochastic Matching with Few Queries: New Algorithms and Tools. In Manuscript. To appear in SODA 2019., 2018.
Soheil Behnezhad and Nima Reyhani. Almost Optimal Stochastic Weighted Matching with Few Queries. In Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, USA, June 18-22, 2018, pages 235-249, 2018.
András A. Benczúr and David R. Karger. Approximating s-t Minimum Cuts in Õ(n^2) Time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 47-55, 1996.
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Aaron Bernstein and Cliff Stein. Fully Dynamic Matching in Bipartite Graphs. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 167-179, 2015.
Aaron Bernstein and Cliff Stein. Faster Fully Dynamic Matchings with Small Approximation Ratios. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 692-711, 2016.
Avrim Blum, John P. Dickerson, Nika Haghtalab, Ariel D. Procaccia, Tuomas Sandholm, and Ankit Sharma. Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, Portland, OR, USA, June 15-19, 2015, pages 325-342, 2015.
Greg Bodwin, Michael Dinitz, Merav Parter, and Virginia Vassilevska Williams. Optimal Vertex Fault Tolerant Spanners (for fixed stretch). In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1884-1900, 2018.
Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving Distances in Very Faulty Graphs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 73:1-73:14, 2017.
Béla Bollobás, Don Coppersmith, and Michael Elkin. Sparse distance preservers and additive spanners. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 414-423, 2003.
Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. Fault-tolerant spanners for general graphs. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 435-444, 2009.
Don Coppersmith and Michael Elkin. Sparse Sourcewise and Pairwise Distance Preservers. SIAM J. Discrete Math., 20(2):463-501, 2006.
Paul Erdős and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. In COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 10. INFINITE AND FINITE SETS, KESZTHELY (HUNGARY). Citeseer, 1973.
Wai Shing Fung, Ramesh Hariharan, Nicholas J. A. Harvey, and Debmalya Panigrahi. A general framework for graph sparsification. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 71-80, 2011.
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the Communication and Streaming Complexity of Maximum Bipartite Matching. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, pages 468-485. SIAM, 2012. URL: http://dl.acm.org/citation.cfm?id=2095116.2095157.
http://dl.acm.org/citation.cfm?id=2095116.2095157
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Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1679-1697, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.121.
http://dx.doi.org/10.1137/1.9781611973105.121
David R. Karger. Random sampling in cut, flow, and network design problems. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 648-657, 1994.
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A New Application of Orthogonal Range Searching for Computing Giant Graph Diameters
A well-known problem for which it is difficult to improve the textbook algorithm is computing the graph diameter. We present two versions of a simple algorithm (one being Monte Carlo and the other deterministic) that for every fixed h and unweighted undirected graph G with n vertices and m edges, either correctly concludes that diam(G) < hn or outputs diam(G), in time O(m+n^{1+o(1)}). The algorithm combines a simple randomized strategy for this problem (Damaschke, IWOCA'16) with a popular framework for computing graph distances that is based on range trees (Cabello and Knauer, Computational Geometry'09). We also prove that under the Strong Exponential Time Hypothesis (SETH), we cannot compute the diameter of a given n-vertex graph in truly subquadratic time, even if the diameter is an Theta(n/log{n}).
Graph diameter
Orthogonal Range Queries
Hardness in P
FPT in P
12:1-12:7
Regular Paper
Guillaume
Ducoffe
Guillaume Ducoffe
10.4230/OASIcs.SOSA.2019.12
A. Abboud, V. Vassilevska Williams, and J. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In SODA, pages 377-391. SIAM, 2016. URL: https://arxiv.org/abs/1506.01799.
https://arxiv.org/abs/1506.01799
J. Bentley and J. Friedman. Data structures for range searching. ACM Computing Surveys (CSUR), 11(4):397-409, 1979.
P. Berman and S. Kasiviswanathan. Faster approximation of distances in graphs. In WADS, pages 541-552. Springer, 2007.
B. Block and M. Milakovic. Computing Diameters in Slim Graphs. Master’s thesis, Chalmers University of Technology, University of Gothenburg, Sweden, 2018. URL: http://publications.lib.chalmers.se/records/fulltext/255208/255208.pdf.
http://publications.lib.chalmers.se/records/fulltext/255208/255208.pdf
J. A. Bondy and U. S. R. Murty. Graph theory. Grad. Texts in Math., 2008.
M. Borassi, P. Crescenzi, and M. Habib. Into the square: On the complexity of some quadratic-time solvable problems. Electronic Notes in Theoretical Computer Science, 322:51-67, 2016. URL: https://arxiv.org/abs/1407.4972.
https://arxiv.org/abs/1407.4972
K. Bringmann, T. Husfeldt, and M. Magnusson. Multivariate Analysis of Orthogonal Range Searching and Graph Distances Parameterized by Treewidth. In IPEC. LIPIcs, 2018. to appear. URL: https://arxiv.org/abs/1805.07135.
https://arxiv.org/abs/1805.07135
S. Cabello and C. Knauer. Algorithms for graphs of bounded treewidth via orthogonal range searching. Computational Geometry, 42(9):815-824, 2009.
T. Chan. Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back. In 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1-27:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. URL: http://drops.dagstuhl.de/opus/volltexte/2017/7226.
http://drops.dagstuhl.de/opus/volltexte/2017/7226
S. Chechik, D. Larkin, L. Roditty, G. Schoenebeck, R. Tarjan, and V. Vassilevska Williams. Better approximation algorithms for the graph diameter. In SODA, pages 1041-1052. SIAM, 2014.
P. Damaschke. Computing Giant Graph Diameters. In IWOCA, pages 373-384. Springer, 2016.
C. Jordan. Sur les assemblages de lignes. J. Reine Angew. Math, 70(185):81, 1869.
A. Meir and J. Moon. Relations between packing and covering numbers of a tree. Pacific Journal of Mathematics, 61(1):225-233, 1975.
L. Monier. Combinatorial solutions of multidimensional divide-and-conquer recurrences. Journal of Algorithms, 1(1):60-74, 1980.
L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In STOC, pages 515-524. ACM, 2013.
Creative Commons Attribution 3.0 Unported license
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Simplified and Space-Optimal Semi-Streaming (2+epsilon)-Approximate Matching
In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a single-pass (2+epsilon)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. Their algorithm uses O(n log^2 n) bits of space, for any constant epsilon>0.
We present a simplified and more intuitive primal-dual analysis, for essentially the same algorithm, which also improves the space complexity to the optimal bound of O(n log n) bits - this is optimal as the output matching requires Omega(n log n) bits.
Streaming
Semi-Streaming
Space-Optimal
Matching
13:1-13:8
Regular Paper
Mohsen
Ghaffari
Mohsen Ghaffari
David
Wajc
David Wajc
10.4230/OASIcs.SOSA.2019.13
Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. Journal of the ACM (JACM), 48(5):1069-1090, 2001.
Reuven Bar-Yehuda, Keren Bendel, Ari Freund, and Dror Rawitz. Local ratio: A unified framework for approximation algorithms. in memoriam: Shimon Even 1935-2004. ACM Computing Surveys (CSUR), 36(4):422-463, 2004.
Michael Crouch and Daniel M Stubbs. Improved Streaming Algorithms for Weighted Matching, via Unweighted Matching. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 96-104, 2014.
Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17(3):449-467, 1965.
Leah Epstein, Asaf Levin, Julián Mestre, and Danny Segev. Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM Journal on Discrete Mathematics, 25(3):1251-1265, 2011.
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Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph
The maximum genus gamma_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least gamma_M(G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong's combinatorial characterisation of maximum genus.
maximum genus
embedding
graph
greedy algorithm
14:1-14:9
Regular Paper
Michal
Kotrbcík
Michal Kotrbcík
Martin
Skoviera
Martin Skoviera
10.4230/OASIcs.SOSA.2019.14
D. Archdeacon, C. P. Bonnington, and J. Širáň. A Nebeský-Type Characterization for Relative Maximum Genus. J. Combin. Theory Ser. B, 73(1):77-98, 1998.
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A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation
In Maximum k-Vertex Cover (Max k-VC), the input is an edge-weighted graph G and an integer k, and the goal is to find a subset S of k vertices that maximizes the total weight of edges covered by S. Here we say that an edge is covered by S iff at least one of its endpoints lies in S.
We present an FPT approximation scheme (FPT-AS) that runs in (1/epsilon)^{O(k)} poly(n) time for the problem, which improves upon Gupta, Lee and Li's (k/epsilon)^{O(k)} poly(n)-time FPT-AS [Anupam Gupta and, 2018; Anupam Gupta et al., 2018]. Our algorithm is simple: just use brute force to find the best k-vertex subset among the O(k/epsilon) vertices with maximum weighted degrees.
Our algorithm naturally yields an (efficient) approximate kernelization scheme of O(k/epsilon) vertices; previously, an O(k^5/epsilon^2)-vertex approximate kernel is only known for the unweighted version of Max k-VC [Daniel Lokshtanov and, 2017]. Interestingly, this also has an application outside of parameterized complexity: using our approximate kernelization as a preprocessing step, we can directly apply Raghavendra and Tan's SDP-based algorithm for 2SAT with cardinality constraint [Prasad Raghavendra and, 2012] to give an 0.92-approximation algorithm for Max k-VC in polynomial time. This improves upon the best known polynomial time approximation algorithm of Feige and Langberg [Uriel Feige and, 2001] which yields (0.75 + delta)-approximation for some (small and unspecified) constant delta > 0.
We also consider the minimization version of the problem (called Min k-VC), where the goal is to find a set S of k vertices that minimizes the total weight of edges covered by S. We provide a FPT-AS for Min k-VC with similar running time of (1/epsilon)^{O(k)} poly(n). Once again, this improves on a (k/epsilon)^{O(k)} poly(n)-time FPT-AS of Gupta et al. On the other hand, we show, assuming a variant of the Small Set Expansion Hypothesis [Raghavendra and Steurer, 2010] and NP !subseteq coNP/poly, that there is no polynomial size approximate kernelization for Min k-VC for any factor less than two.
Maximum k-Vertex Cover
Minimum k-Vertex Cover
Approximation Algorithms
Fixed Parameter Algorithms
Approximate Kernelization
15:1-15:21
Regular Paper
Pasin
Manurangsi
Pasin Manurangsi
10.4230/OASIcs.SOSA.2019.15
Faisal N. Abu-Khzam, Rebecca L. Collins, Michael R. Fellows, Michael A. Langston, W. Henry Suters, and Christopher T. Symons. Kernelization algorithms for the vertex cover problem: Theory and experiments. In ALENEX, pages 62-69, 2004.
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Irit Dinur. Mildly exponential reduction from gap 3sat to polynomial-gap label-cover. Electronic Colloquium on Computational Complexity (ECCC), 23:128, 2016.
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Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On non-optimally expanding sets in grassmann graphs. ECCC, 24:94, 2017.
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Anupam Gupta, Euiwoong Lee, and Jason Li. Faster exact and approximate algorithms for k-cut. In FOCS, 2018. To appear.
Anupam Gupta, Euiwoong Lee, and Jason Li. An FPT algorithm beating 2-approximation for k-cut. In SODA, pages 2821-2837, 2018.
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Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and grassmann graphs. In STOC, pages 576-589, 2017.
Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. ECCC, 25:6, 2018.
Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008.
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Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In STOC, pages 224-237, 2017.
Pasin Manurangsi. A note on max k-vertex cover: Faster fpt-as, smaller approximate kernel and improved approximation. arXiv preprint arXiv:1810.03792, 2018.
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Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008.
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Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Simple Contention Resolution via Multiplicative Weight Updates
We consider the classic contention resolution problem, in which devices conspire to share some common resource, for which they each need temporary and exclusive access. To ground the discussion, suppose (identical) devices wake up at various times, and must send a single packet over a shared multiple-access channel. In each time step they may attempt to send their packet; they receive ternary feedback {0,1,2^+} from the channel, 0 indicating silence (no one attempted transmission), 1 indicating success (one device successfully transmitted), and 2^+ indicating noise. We prove that a simple strategy suffices to achieve a channel utilization rate of 1/e-O(epsilon), for any epsilon>0. In each step, device i attempts to send its packet with probability p_i, then applies a rudimentary multiplicative weight-type update to p_i.
p_i <- { p_i * e^{epsilon} upon hearing silence (0), p_i upon hearing success (1), p_i * e^{-epsilon/(e-2)} upon hearing noise (2^+) }.
This scheme works well even if the introduction of devices/packets is adversarial, and even if the adversary can jam time slots (make noise) at will. We prove that if the adversary jams J time slots, then this scheme will achieve channel utilization 1/e-epsilon, excluding O(J) wasted slots. Results similar to these (Bender, Fineman, Gilbert, Young, SODA 2016) were already achieved, but with a lower constant efficiency (less than 0.05) and a more complex algorithm.
Contention resolution
multiplicative weight update method
16:1-16:16
Regular Paper
Yi-Jun
Chang
Yi-Jun Chang
Wenyu
Jin
Wenyu Jin
Seth
Pettie
Seth Pettie
10.4230/OASIcs.SOSA.2019.16
D. J. Aldous. Ultimate instability of exponential back-off protocol for acknowledgment-based transmission control of random access communication channels. IEEE Trans. Information Theory, 33(2):219-223, 1987. URL: http://dx.doi.org/10.1109/TIT.1987.1057295.
http://dx.doi.org/10.1109/TIT.1987.1057295
S. Arora, E. Hazan, and S. Kale. The Multiplicative Weights Update Method: a Meta-Algorithm and Applications. Theory of Computing, 8(1):121-164, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a006.
http://dx.doi.org/10.4086/toc.2012.v008a006
B. Awerbuch, A. W. Richa, and C. Scheideler. A jamming-resistant MAC protocol for single-hop wireless networks. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 45-54, 2008. URL: http://dx.doi.org/10.1145/1400751.1400759.
http://dx.doi.org/10.1145/1400751.1400759
M. A. Bender, M. Farach-Colton, S. He, B. C. Kuszmaul, and C. E. Leiserson. Adversarial contention resolution for simple channels. In Proceedings of the 17th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 325-332, 2005. URL: http://dx.doi.org/10.1145/1073970.1074023.
http://dx.doi.org/10.1145/1073970.1074023
M. A. Bender, J. T. Fineman, and S. Gilbert. Contention Resolution with Heterogeneous Job Sizes. In Proceedings 14th Annual European Symposium on Algorithms (ESA), pages 112-123, 2006. URL: http://dx.doi.org/10.1007/11841036_13.
http://dx.doi.org/10.1007/11841036_13
M. A. Bender, J. T. Fineman, S. Gilbert, and M. Young. How to Scale Exponential Backoff: Constant Throughput, Polylog Access Attempts, and Robustness. In Proceedings 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 636-654, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch47.
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M. Herlihy and J. E. B. Moss. Transactional Memory: Architectural Support for Lock-Free Data Structures. In Proceedings of the 20th Annual International Symposium on Computer Architecture (ISCA), pages 289-300, 1993. URL: http://dx.doi.org/10.1145/165123.165164.
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http://dx.doi.org/10.1145/52324.52356
J. F. Kurose and K. W. Ross. Computer networking: a top-down approach, volume 4. Addison-Wesley, Boston, 2009.
K. Li, I. Nikolaidis, and J. J. Harms. The analysis of the additive-increase multiplicative-decrease MAC protocol. In Proceedings 10th Annual Conference on Wireless On-demand Network Systems and Services (WONS), pages 122-124, 2013. URL: http://dx.doi.org/10.1109/WONS.2013.6578335.
http://dx.doi.org/10.1109/WONS.2013.6578335
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V. A. Mikhailov and B. S. Tsybakov. Upper bound for the capacity of a random multiple access system. Problemy Peredachi Informatsii, 17(1):90-95, 1981.
A. Mondal and A. Kuzmanovic. Removing exponential backoff from TCP. Computer Communication Review, 38(5):17-28, 2008. URL: http://dx.doi.org/10.1145/1452335.1452338.
http://dx.doi.org/10.1145/1452335.1452338
J. Mosely and P. A. Humblet. A Class of Efficient Contention Resolution Algorithms for Multiple Access Channels. IEEE Trans. Communications, 33(2):145-151, 1985. URL: http://dx.doi.org/10.1109/TCOM.1985.1096261.
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R. Rajwar and J. R. Goodman. Speculative lock elision: enabling highly concurrent multithreaded execution. In Proceedings of the 34th Annual International Symposium on Microarchitecture (MICRO), pages 294-305, 2001. URL: http://dx.doi.org/10.1109/MICRO.2001.991127.
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N.-O. Song, B.-J. Kwak, and L. E. Miller. Analysis of EIED backoff algorithm for the IEEE 802.11 DCF. In Proceedings 62nd IEEE Vehicular Technology Conference (VTC), volume 4, pages 2182-2186, 2005.
B. S. Tsybakov and V. A. Mikhailov. Slotted multiaccess packet broadcasting feedback channel. Problemy Peredachi Informatsii, 14(4):32-59, 1978.
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A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum
Given a multiset S of n positive integers and a target integer t, the Subset Sum problem asks to determine whether there exists a subset of S that sums up to t. The current best deterministic algorithm, by Koiliaris and Xu [SODA'17], runs in O~(sqrt{n}t) time, where O~ hides poly-logarithm factors. Bringmann [SODA'17] later gave a randomized O~(n + t) time algorithm using two-stage color-coding. The O~(n+t) running time is believed to be near-optimal.
In this paper, we present a simple and elegant randomized algorithm for Subset Sum in O~(n + t) time. Our new algorithm actually solves its counting version modulo prime p>t, by manipulating generating functions using FFT.
subset sum
formal power series
FFT
17:1-17:6
Regular Paper
Ce
Jin
Ce Jin
Hongxun
Wu
Hongxun Wu
10.4230/OASIcs.SOSA.2019.17
Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-based lower bounds for subset sum and bicriteria path. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2019. To appear. URL: http://arxiv.org/abs/1704.04546.
http://arxiv.org/abs/1704.04546
Richard E. Bellman. Dynamic programming. Princeton University Press, 1957.
Richard P. Brent. Multiple-precision zero-finding methods and the complexity of elementary function evaluation. In Analytic Computational Complexity, pages 151-176. Elsevier, 1976. URL: http://dx.doi.org/10.1016/B978-0-12-697560-4.50014-9.
http://dx.doi.org/10.1016/B978-0-12-697560-4.50014-9
Karl Bringmann. A near-linear pseudopolynomial time algorithm for subset sum. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1073-1084, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.69.
http://dx.doi.org/10.1137/1.9781611974782.69
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 3rd edition, 2009.
Martin Dyer. Approximate counting by dynamic programming. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages 693-699, 2003. URL: http://dx.doi.org/10.1145/780542.780643.
http://dx.doi.org/10.1145/780542.780643
Paweł Gawrychowski, Liran Markin, and Oren Weimann. A Faster FPTAS for #Knapsack. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP), pages 64:1-64:13, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.64.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.64
Parikshit Gopalan, Adam Klivans, Raghu Meka, Daniel Štefankovic, Santosh Vempala, and Eric Vigoda. An FPTAS for #knapsack and related counting problems. In Proceedings of the 52nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 817-826, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.32.
http://dx.doi.org/10.1109/FOCS.2011.32
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http://dx.doi.org/10.1007/978-1-4684-2001-2_9
Konstantinos Koiliaris and Chao Xu. A faster pseudopolynomial time algorithm for subset sum. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1062-1072, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.68.
http://dx.doi.org/10.1137/1.9781611974782.68
Konstantinos Koiliaris and Chao Xu. Subset Sum Made Simple. CoRR, abs/1807.08248, 2018. URL: http://arxiv.org/abs/1807.08248.
http://arxiv.org/abs/1807.08248
David Pisinger. Linear time algorithms for knapsack problems with bounded weights. Journal of Algorithms, 33(1):1-14, 1999. URL: http://dx.doi.org/10.1006/jagm.1999.1034.
http://dx.doi.org/10.1006/jagm.1999.1034
Romeo Rizzi and Alexandru I. Tomescu. Faster FPTASes for counting and random generation of knapsack solutions. In European Symposium on Algorithms (ESA), pages 762-773, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_63.
http://dx.doi.org/10.1007/978-3-662-44777-2_63
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Submodular Optimization in the MapReduce Model
Submodular optimization has received significant attention in both practice and theory, as a wide array of problems in machine learning, auction theory, and combinatorial optimization have submodular structure. In practice, these problems often involve large amounts of data, and must be solved in a distributed way. One popular framework for running such distributed algorithms is MapReduce. In this paper, we present two simple algorithms for cardinality constrained submodular optimization in the MapReduce model: the first is a (1/2-o(1))-approximation in 2 MapReduce rounds, and the second is a (1-1/e-epsilon)-approximation in (1+o(1))/epsilon MapReduce rounds.
mapreduce
submodular
optimization
approximation algorithms
18:1-18:10
Regular Paper
Paul
Liu
Paul Liu
Jan
Vondrak
Jan Vondrak
10.4230/OASIcs.SOSA.2019.18
Sepehr Assadi and Sanjeev Khanna. Randomized Composable Coresets for Matching and Vertex Cover. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2017.
Sepehr Assadi and Sanjeev Khanna. Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2412-2431, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.155.
http://dx.doi.org/10.1137/1.9781611975031.155
Eric Balkanski, Adam Breuer, and Yaron Singer. Non-monotone Submodular Maximization in Exponentially Fewer Iterations. CoRR, abs/1807.11462, 2018. URL: http://arxiv.org/abs/1807.11462.
http://arxiv.org/abs/1807.11462
Eric Balkanski, Aviad Rubinstein, and Yaron Singer. An Exponential Speedup in Parallel Running Time for Submodular Maximization without Loss in Approximation. CoRR, abs/1804.06355, 2018. URL: http://arxiv.org/abs/1804.06355.
http://arxiv.org/abs/1804.06355
Rafael da Ponte Barbosa, Alina Ene, Huy L. Nguyen, and Justin Ward. A New Framework for Distributed Submodular Maximization. In Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, 2016.
Satoru Fujishige. Submodular functions and optimization, volume 58. Elsevier, 2005.
Howard J. Karloff, Siddharth Suri, and Sergei Vassilvitskii. A Model of Computation for MapReduce. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 938-948, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.76.
http://dx.doi.org/10.1137/1.9781611973075.76
Ravi Kumar, Benjamin Moseley, Sergei Vassilvitskii, and Andrea Vattani. Fast Greedy Algorithms in MapReduce and Streaming. TOPC, 2(3):14:1-14:22, 2015. URL: http://dx.doi.org/10.1145/2809814.
http://dx.doi.org/10.1145/2809814
Andrew McGregor and Hoa T. Vu. Better Streaming Algorithms for the Maximum Coverage Problem. In 20th International Conference on Database Theory, ICDT 2017, March 21-24, 2017, Venice, Italy, pages 22:1-22:18, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICDT.2017.22.
http://dx.doi.org/10.4230/LIPIcs.ICDT.2017.22
Vahab S. Mirrokni and Morteza Zadimoghaddam. Randomized Composable Core-sets for Distributed Submodular Maximization. In ACM Symposium on Theory of Computing (STOC), pages 153-162, 2015.
George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Program., 14(1):265-294, 1978. URL: http://dx.doi.org/10.1007/BF01588971.
http://dx.doi.org/10.1007/BF01588971
Martin Raab and Angelika Steger. "Balls into Bins" - A Simple and Tight Analysis. In Michael Luby, José D. P. Rolim, and Maria Serna, editors, Randomization and Approximation Techniques in Computer Science, pages 159-170, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg.
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Compressed Sensing with Adversarial Sparse Noise via L1 Regression
We present a simple and effective algorithm for the problem of sparse robust linear regression. In this problem, one would like to estimate a sparse vector w^* in R^n from linear measurements corrupted by sparse noise that can arbitrarily change an adversarially chosen eta fraction of measured responses y, as well as introduce bounded norm noise to the responses.
For Gaussian measurements, we show that a simple algorithm based on L1 regression can successfully estimate w^* for any eta < eta_0 ~~ 0.239, and that this threshold is tight for the algorithm. The number of measurements required by the algorithm is O(k log n/k) for k-sparse estimation, which is within constant factors of the number needed without any sparse noise.
Of the three properties we show - the ability to estimate sparse, as well as dense, w^*; the tolerance of a large constant fraction of outliers; and tolerance of adversarial rather than distributional (e.g., Gaussian) dense noise - to the best of our knowledge, no previous polynomial time algorithm was known to achieve more than two.
Robust Regression
Compressed Sensing
19:1-19:19
Regular Paper
Sushrut
Karmalkar
Sushrut Karmalkar
Eric
Price
Eric Price
10.4230/OASIcs.SOSA.2019.19
Kush Bhatia, Prateek Jain, Parameswaran Kamalaruban, and Purushottam Kar. Consistent Robust Regression. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 2110-2119. Curran Associates, Inc., 2017. URL: http://papers.nips.cc/paper/6806-consistent-robust-regression.pdf.
http://papers.nips.cc/paper/6806-consistent-robust-regression.pdf
Kush Bhatia, Prateek Jain, and Purushottam Kar. Robust regression via hard thresholding. In Advances in Neural Information Processing Systems, pages 721-729, 2015.
P. Bloomfield and W. Steiger. Least Absolute Deviations Curve-Fitting. SIAM Journal on Scientific and Statistical Computing, 1(2):290-301, 1980. URL: http://dx.doi.org/10.1137/0901019.
http://dx.doi.org/10.1137/0901019
E. J. Candès, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59(8):1208-1223, 2006.
I. Diakonikolas, W. Kong, and A. Stewart. Efficient Algorithms and Lower Bounds for Robust Linear Regression. ArXiv e-prints, May 2018. URL: http://arxiv.org/abs/1806.00040.
http://arxiv.org/abs/1806.00040
Ilias Diakonikolas, Gautam Kamath, Daniel M. Kane, Jerry Li, Jacob Steinhardt, and Alistair Stewart. Sever: A Robust Meta-Algorithm for Stochastic Optimization. CoRR, abs/1803.02815, 2018. URL: http://arxiv.org/abs/1803.02815.
http://arxiv.org/abs/1803.02815
Cynthia Dwork, Frank McSherry, and Kunal Talwar. The price of privacy and the limits of LP decoding. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 85-94. ACM, 2007.
Rina Foygel and Lester Mackey. Corrupted sensing: Novel guarantees for separating structured signals. IEEE Transactions on Information Theory, 60(2):1223-1247, 2014.
Peter J Huber. Robust statistics. In International Encyclopedia of Statistical Science, pages 1248-1251. Springer, 2011.
Adam R. Klivans, Pravesh K. Kothari, and Raghu Meka. Efficient Algorithms for Outlier-Robust Regression. In Conference On Learning Theory, COLT 2018, Stockholm, Sweden, 6-9 July 2018., pages 1420-1430, 2018. URL: http://proceedings.mlr.press/v75/klivans18a.html.
http://proceedings.mlr.press/v75/klivans18a.html
Jason N Laska, Mark A Davenport, and Richard G Baraniuk. Exact signal recovery from sparsely corrupted measurements through the pursuit of justice. In Signals, Systems and Computers, 2009 Conference Record of the Forty-Third Asilomar Conference on, pages 1556-1560. IEEE, 2009.
Xiaodong Li. Compressed sensing and matrix completion with constant proportion of corruptions. Constructive Approximation, 37(1):73-99, 2013.
Liu Liu, Yanyao Shen, Tianyang Li, and Constantine Caramanis. High Dimensional Robust Sparse Regression. arXiv preprint arXiv:1805.11643, 2018.
Nasser M Nasrabadi, Trac D Tran, and Nam Nguyen. Robust lasso with missing and grossly corrupted observations. In Advances in Neural Information Processing Systems, pages 1881-1889, 2011.
Nam H Nguyen and Trac D Tran. Exact Recoverability From Dense Corrupted Observations via L1-Minimization. IEEE transactions on information theory, 59(4):2017-2035, 2013.
Hansheng Wang, Guodong Li, and Guohua Jiang. Robust Regression Shrinkage and Consistent Variable Selection through the LAD-Lasso. Journal of Business &Economic Statistics, 25(3):347-355, 2007. URL: http://www.jstor.org/stable/27638939.
http://www.jstor.org/stable/27638939
Meng Wang, Weiyu Xu, and Ao Tang. The Limits of Error Correction with lp Decoding. CoRR, abs/1006.0277, 2010.
Chun Yu and Weixin Yao. Robust linear regression: A review and comparison. Communications in Statistics-Simulation and Computation, 46(8):6261-6282, 2017.
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Approximating Maximin Share Allocations
We study the problem of fair allocation of M indivisible items among N agents using the popular notion of maximin share as our measure of fairness. The maximin share of an agent is the largest value she can guarantee herself if she is allowed to choose a partition of the items into N bundles (one for each agent), on the condition that she receives her least preferred bundle. A maximin share allocation provides each agent a bundle worth at least their maximin share. While it is known that such an allocation need not exist [Procaccia and Wang, 2014; Kurokawa et al., 2016], a series of work [Procaccia and Wang, 2014; David Kurokawa et al., 2018; Amanatidis et al., 2017; Barman and Krishna Murthy, 2017] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times their maximin share. Recently, [Ghodsi et al., 2018] improved the approximation guarantee to 3/4. Prior works utilize intricate algorithms, with an exception of [Barman and Krishna Murthy, 2017] which is a simple greedy solution but relies on sophisticated analysis techniques. In this paper, we propose an alternative 2/3 maximin share approximation which offers both a simple algorithm and straightforward analysis. In contrast to other algorithms, our approach allows for a simple and intuitive understanding of why it works.
Fair division
Maximin share
Approximation algorithm
20:1-20:11
Regular Paper
Jugal
Garg
Jugal Garg
Peter
McGlaughlin
Peter McGlaughlin
Setareh
Taki
Setareh Taki
10.4230/OASIcs.SOSA.2019.20
Georgios Amanatidis, Evangelos Markakis, Afshin Nikzad, and Amin Saberi. Approximation algorithms for computing maximin share allocations. ACM Transactions on Algorithms (TALG), 13(4):52, 2017.
Siddharth Barman and Sanath Kumar Krishna Murthy. Approximation algorithms for maximin fair division. In Proceedings of the 2017 ACM Conference on Economics and Computation, pages 647-664. ACM, 2017.
Sylvain Bouveret and Michel Lemaître. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259-290, 2016.
Sylvain Bouveret and Michel Lemaître. Efficiency and sequenceability in fair division of indivisible goods with additive preferences. arXiv preprint arXiv:1604.01734, 2016.
Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6):1061-1103, 2011.
Mohammad Ghodsi, MohammadTaghi HajiAghayi, Masoud Seddighin, Saeed Seddighin, and Hadi Yami. Fair allocation of indivisible goods: Improvement and generalization. In EC, 2018.
David Kurokawa, Ariel D Procaccia, and Junxing Wang. When can the maximin share guarantee be guaranteed? In AAAI, volume 16, pages 523-529, 2016.
David Kurokawa, Ariel D. Procaccia, and Junxing Wang. Fair Enough: Guaranteeing Approximate Maximin Shares. J. ACM, 65(2):8:1-8:27, 2018.
Ariel D Procaccia and Junxing Wang. Fair enough: Guaranteeing approximate maximin shares. In Proceedings of the fifteenth ACM conference on Economics and computation, pages 675-692. ACM, 2014.
Hugo Steinhaus. The problem of fair division. Econometrica, 16:101-104, 1948.
Gerhard J Woeginger. A polynomial-time approximation scheme for maximizing the minimum machine completion time. Operations Research Letters, 20(4):149-154, 1997.
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