Tight Approximation Guarantees for Concave Coverage Problems

Authors Siddharth Barman, Omar Fawzi, Paul Fermé

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Author Details

Siddharth Barman
  • Indian Institute of Science, Bangalore, India
Omar Fawzi
  • Univ. Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France
Paul Fermé
  • Univ. Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France

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Siddharth Barman, Omar Fawzi, and Paul Fermé. Tight Approximation Guarantees for Concave Coverage Problems. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In the maximum coverage problem, we are given subsets T_1, …, T_m of a universe [n] along with an integer k and the objective is to find a subset S ⊆ [m] of size k that maximizes C(S) : = |⋃_{i ∈ S} T_i|. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of 1-e^{-1}. In this work we consider a generalization of this problem wherein an element a can contribute by an amount that depends on the number of times it is covered. Given a concave, nondecreasing function φ, we define C^{φ}(S) := ∑_{a ∈ [n]}w_aφ(|S|_a), where |S|_a = |{i ∈ S : a ∈ T_i}|. The standard maximum coverage problem corresponds to taking φ(j) = min{j,1}. For any such φ, we provide an efficient algorithm that achieves an approximation ratio equal to the Poisson concavity ratio of φ, defined by α_{φ} : = min_{x ∈ ℕ^*} 𝔼[φ(Poi(x))] / φ(𝔼[Poi(x)]). Complementing this approximation guarantee, we establish a matching NP-hardness result when φ grows in a sublinear way. As special cases, we improve the result of [Siddharth Barman et al., 2020] about maximum multi-coverage, that was based on the unique games conjecture, and we recover the result of [Szymon Dudycz et al., 2020] on multi-winner approval-based voting for geometrically dominant rules. Our result goes beyond these special cases and we illustrate it with applications to distributed resource allocation problems, welfare maximization problems and approval-based voting for general rules.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Rounding techniques
  • Theory of computation → Algorithmic game theory
  • Approximation Algorithms
  • Coverage Problems
  • Concave Function


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  1. Alexander A. Ageev and Maxim Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. Comb. Optim., 8(3):307-328, 2004. URL: https://doi.org/10.1023/B:JOCO.0000038913.96607.c2.
  2. Siddharth Barman and Omar Fawzi. Algorithmic aspects of optimal channel coding. IEEE Trans. Inf. Theory, 64(2):1038-1045, 2018. URL: https://doi.org/10.1109/TIT.2017.2696963.
  3. Siddharth Barman, Omar Fawzi, and Paul Fermé. Tight approximation guarantees for concave coverage problems. CoRR, abs/2010.00970, 2020. URL: http://arxiv.org/abs/2010.00970.
  4. Siddharth Barman, Omar Fawzi, Suprovat Ghoshal, and Emirhan Gürpinar. Tight approximation bounds for maximum multi-coverage. In Daniel Bienstock and Giacomo Zambelli, editors, Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, London, UK, June 8-10, 2020, Proceedings, volume 12125 of Lecture Notes in Computer Science, pages 66-77. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-45771-6_6.
  5. Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors. Handbook of Computational Social Choice. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781107446984.
  6. Markus Brill, Jean-François Laslier, and Piotr Skowron. Multiwinner approval rules as apportionment methods. In Satinder P. Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4-9, 2017, San Francisco, California, USA, pages 414-420. AAAI Press, 2017. URL: https://doi.org/10.1177/0951629818775518.
  7. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  8. Rahul Chandan, Dario Paccagnan, and Jason R. Marden. Optimal mechanisms for distributed resource-allocation. CoRR, abs/1911.07823, 2019. URL: http://arxiv.org/abs/1911.07823.
  9. Michele Conforti and Gérard Cornuéjols. Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discret. Appl. Math., 7(3):251-274, 1984. URL: https://doi.org/10.1016/0166-218X(84)90003-9.
  10. Gérard Cornuéjols, Marshall L Fisher, and George L Nemhauser. Exceptional paper—location of bank accounts to optimize float: An analytic study of exact and approximate algorithms. Management science, 23(8):789-810, 1977. URL: https://doi.org/10.1287/mnsc.23.8.789.
  11. Szymon Dudycz, Pasin Manurangsi, Jan Marcinkowski, and Krzysztof Sornat. Tight approximation for proportional approval voting. In Christian Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020, pages 276-282. ijcai.org, 2020. URL: https://doi.org/10.24963/ijcai.2020/39.
  12. Shaddin Dughmi and Jan Vondrák. Limitations of randomized mechanisms for combinatorial auctions. Games Econ. Behav., 92:370-400, 2015. URL: https://doi.org/10.1016/j.geb.2014.01.007.
  13. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  14. Vitaly Feldman and Pravesh Kothari. Learning coverage functions and private release of marginals. In Maria-Florina Balcan, Vitaly Feldman, and Csaba Szepesvári, editors, Proceedings of The 27th Conference on Learning Theory, COLT 2014, Barcelona, Spain, June 13-15, 2014, volume 35 of JMLR Workshop and Conference Proceedings, pages 679-702. JMLR.org, 2014. URL: http://proceedings.mlr.press/v35/feldman14a.html.
  15. Dorit S. Hochbaum. Approximation algorithms for NP-hard problems. SIGACT News, 28(2):40-52, 1997. URL: https://doi.org/10.1145/261342.571216.
  16. Jason R. Marden and Adam Wierman. Distributed welfare games with applications to sensor coverage. In Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, December 9-11, 2008, Cancún, Mexico, pages 1708-1713. IEEE, 2008. URL: https://doi.org/10.1109/CDC.2008.4738800.
  17. Robert A Murphey. Target-based weapon target assignment problems. In Nonlinear assignment problems, pages 39-53. Springer, 2000. URL: https://doi.org/10.1007/978-1-4757-3155-2_3.
  18. Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511800481.
  19. Dario Paccagnan and Jason R Marden. Utility design for distributed resource allocation-part II: Applications to submodular, covering, and supermodular problems. CoRR, abs/1807.01343, 2018. URL: http://arxiv.org/abs/1807.01343.
  20. Moshe Shaked and J George Shanthikumar. Stochastic orders. Springer Science & Business Media, 2007. URL: https://doi.org/10.1007/978-0-387-34675-5.
  21. Piotr Skowron, Piotr Faliszewski, and Jérôme Lang. Finding a collective set of items: From proportional multirepresentation to group recommendation. Artif. Intell., 241:191-216, 2016. URL: https://doi.org/10.1016/j.artint.2016.09.003.
  22. Maxim Sviridenko, Jan Vondrák, and Justin Ward. Optimal approximation for submodular and supermodular optimization with bounded curvature. Math. Oper. Res., 42(4):1197-1218, 2017. URL: https://doi.org/10.1287/moor.2016.0842.
  23. Jan Vondrák. Submodularity in Combinatorial Optimization. Univerzita Karlova, Matematicko-Fyzikální Fakulta, 2007. URL: https://dspace.cuni.cz/bitstream/handle/20.500.11956/13738/140038775.pdf.