Document Open Access Logo

A Tight Competitive Ratio for Online Submodular Welfare Maximization

Authors Amit Ganz, Pranav Nuti, Roy Schwartz

PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.52.pdf
  • Filesize: 0.72 MB
  • 17 pages

Document Identifiers

Author Details

Amit Ganz
  • The Henry and Marilyn Taub Faculty of Computer Science, Technion, Haifa, Israel
Pranav Nuti
  • Department of Computer Science, Stanford University, CA, USA
Roy Schwartz
  • The Henry and Marilyn Taub Faculty of Computer Science, Technion, Haifa, Israel

Funding

Amit Ganz and Roy Schwartz have received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 852870-ERC-SUBMODULAR.

Acknowledgements

The authors would like to thank the anonymous referees for helpful remarks.

Cite AsGet BibTex

Amit Ganz, Pranav Nuti, and Roy Schwartz. A Tight Competitive Ratio for Online Submodular Welfare Maximization. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 52:1-52:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.52

Abstract

In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given n bidders each equipped with a general non-negative (not necessarily monotone) submodular utility and m items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of 1/4. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of 3-2√2≈ 0.171573 to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of ≈ 0.27493, improving the previously known 1/4 guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Algorithms
  • Submodular Maximization
  • Welfare Maximization
  • Approximation Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online Vertex-Weighted Bipartite Matching and Single-bid Budgeted Allocations, pages 1253-1264. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.95.
  2. Shipra Agrawal, Zizhuo Wang, and Yinyu Ye. A dynamic near-optimal algorithm for online linear programming. Operations Research, 62(4):876-890, 2014. Google Scholar
  3. Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '07, pages 434-443, USA, 2007. Society for Industrial and Applied Mathematics. Google Scholar
  4. Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1497-1514, 2014. Google Scholar
  5. Siddharth Barman, Seeun Umboh, Shuchi Chawla, and David Malec. Secretary problems with convex costs. In Artur Czumaj, Kurt Mehlhorn, Andrew Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming, pages 75-87, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  6. Mohammadhossein Bateni, Mohammadtaghi Hajiaghayi, and Morteza Zadimoghaddam. Submodular secretary problem and extensions. ACM Trans. Algorithms, 9(4), October 2013. URL: https://doi.org/10.1145/2500121.
  7. N. Buchbinder, M. Feldman, J. Naor, and R. Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM Journal on Computing, 44(5):1384-1402, 2015. Google Scholar
  8. Niv Buchbinder and Moran Feldman. Constrained submodular maximization via a nonsymmetric technique. Mathematics of Operations Research, 44(3):988-1005, 2019. Google Scholar
  9. Niv Buchbinder, Moran Feldman, Yuval Filmus, and Mohit Garg. Online submodular maximization: Beating 1/2 made simple. Mathematical Programming, 183(1-2):149-169, 2020. Google Scholar
  10. Niv Buchbinder, Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. Submodular maximization with cardinality constraints. In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1433-1452, 2014. Google Scholar
  11. Niv Buchbinder, Moran Feldman, and Roy Schwartz. Online submodular maximization with preemption. ACM Trans. Algorithms, 15(3), June 2019. URL: https://doi.org/10.1145/3309764.
  12. Niv Buchbinder, Kamal Jain, and Joseph (Seffi) Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In Lars Arge, Michael Hoffmann, and Emo Welzl, editors, Algorithms - ESA 2007, pages 253-264, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. Google Scholar
  13. Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a submodular set function subject to a matroid constraint (extended abstract). In Matteo Fischetti and David P. Williamson, editors, Integer Programming and Combinatorial Optimization, pages 182-196, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. Google Scholar
  14. Gruia Calinescu, 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. Google Scholar
  15. Deeparnab Chakrabarty and Gagan Goel. On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and gap. SIAM Journal on Computing, 39(6):2189-2211, 2010. URL: https://doi.org/10.1137/080735503.
  16. Nikhil R. Devanur and Thomas P. Hayes. The adwords problem: Online keyword matching with budgeted bidders under random permutations. In Proceedings of the 10th ACM Conference on Electronic Commerce, EC '09, pages 71-78, New York, NY, USA, 2009. Association for Computing Machinery. URL: https://doi.org/10.1145/1566374.1566384.
  17. Nikhil R Devanur, Zhiyi Huang, Nitish Korula, Vahab S Mirrokni, and Qiqi Yan. Whole-page optimization and submodular welfare maximization with online bidders. ACM Transactions on Economics and Computation (TEAC), 4(3):1-20, 2016. Google Scholar
  18. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctions with complement-free bidders. Mathematics of Operations Research, 35(1):1-13, 2010. URL: https://doi.org/10.1287/moor.1090.0436.
  19. Shahar Dobzinski and Michael Schapira. An improved approximation algorithm for combinatorial auctions with submodular bidders. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA '06, pages 1064-1073, USA, 2006. Society for Industrial and Applied Mathematics. Google Scholar
  20. E. B. Dynkin. The optimum choice of the instant for stopping a markov process, 1963. Google Scholar
  21. Alina Ene and Huy L. Nguyen. Constrained submodular maximization: Beyond 1/e. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 248-257, 2016. URL: https://doi.org/10.1109/FOCS.2016.34.
  22. Alina Ene and Huy L. Nguyen. A nearly-linear time algorithm for submodular maximization with knapsack, partition and graphical matroid constraints. CoRR, abs/1709.09767, 2018. Google Scholar
  23. Uriel Feige. On maximizing welfare when utility functions are subadditive. SIAM Journal on Computing, 39(1):122-142, 2009. URL: https://doi.org/10.1137/070680977.
  24. Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM Journal on Computing, 40(4):1133-1153, 2011. Google Scholar
  25. Uriel Feige and Jan Vondrak. Approximation algorithms for allocation problems: Improving the factor of 1 - 1/e. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 667-676, 2006. URL: https://doi.org/10.1109/FOCS.2006.14.
  26. Uriel Feige and Jan Vondrák. The submodular welfare problem with demand queries. Theory of Computing, 6(11):247-290, 2010. URL: https://doi.org/10.4086/toc.2010.v006a011.
  27. Jon Feldman, Monika Henzinger, Nitish Korula, Vahab S Mirrokni, and Cliff Stein. Online stochastic packing applied to display ad allocation. In European Symposium on Algorithms, pages 182-194. Springer, 2010. Google Scholar
  28. Jon Feldman, Nitish Korula, Vahab Mirrokni, S. Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In Stefano Leonardi, editor, Internet and Network Economics, pages 374-385, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. Google Scholar
  29. Jon Feldman, Aranyak Mehta, Vahab Mirrokni, and S. Muthukrishnan. Online stochastic matching: Beating 1-1/e. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 117-126, 2009. URL: https://doi.org/10.1109/FOCS.2009.72.
  30. Moran Feldman. Maximizing symmetric submodular functions. ACM Transactions on Algorithms (TALG), 13(3):1-36, 2017. Google Scholar
  31. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 570-579, 2011. URL: https://doi.org/10.1109/FOCS.2011.46.
  32. Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. Nonmonotone submodular maximization via a structural continuous greedy algorithm. In ICALP, pages 342-353, 2011. Google Scholar
  33. Moran Feldman, Ola Svensson, and Rico Zenklusen. A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem, pages 1189-1201. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.79.
  34. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1098-1116. SIAM, 2011. Google Scholar
  35. Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, volume 8, pages 982-991. Citeseer, 2008. Google Scholar
  36. Corinna Gottschalk and Britta Peis. Submodular function maximization on the bounded integer lattice. In Approximation and Online Algorithms. Springer International Publishing, 2015. Google Scholar
  37. Anupam Gupta, Aaron Roth, Grant Schoenebeck, and Kunal Talwar. Constrained non-monotone submodular maximization: Offline and secretary algorithms. In Amin Saberi, editor, Internet and Network Economics, pages 246-257, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. Google Scholar
  38. Christopher Harshaw, Ehsan Kazemi, Moran Feldman, and Amin Karbasi. The power of subsampling in submodular maximization. Mathematics of Operations Research, 47(2):1365-1393, 2022. Google Scholar
  39. Xinran He and Yan Liu. Not enough data? joint inferring multiple diffusion networks via network generation priors. In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, WSDM '17, pages 465-474, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3018661.3018675.
  40. Bala Kalyanasundaram and Kirk R. Pruhs. An optimal deterministic algorithm for online b-matching. Theoretical Computer Science, 233(1):319-325, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00140-1.
  41. Mikhail Kapralov, Ian Post, and Jan Vondrák. Online and stochastic variants of welfare maximization. CoRR, abs/1204.1025, 2012. URL: https://arxiv.org/abs/1204.1025.
  42. R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, STOC '90, pages 352-358, New York, NY, USA, 1990. Association for Computing Machinery. URL: https://doi.org/10.1145/100216.100262.
  43. Subhash Khot, Richard J. Lipton, Evangelos Markakis, and Aranyak Mehta. Inapproximability results for combinatorial auctions with submodular utility functions. In Proceedings of the First International Conference on Internet and Network Economics, WINE'05, pages 92-101, 2005. Google Scholar
  44. Samir Khuller, Anna Moss, and Joseph (Seffi) Naor. The budgeted maximum coverage problem. Information Processing Letters, 70(1):39-45, 1999. URL: https://doi.org/10.1016/S0020-0190(99)00031-9.
  45. Robert Kleinberg. A multiple-choice secretary algorithm with applications to online auctions. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '05, pages 630-631, USA, 2005. Society for Industrial and Applied Mathematics. Google Scholar
  46. Nitish Korula, Vahab Mirrokni, and Morteza Zadimoghaddam. Online submodular welfare maximization: Greedy beats 1/2 in random order. SIAM Journal on Computing, 47(3):1056-1086, 2018. URL: https://doi.org/10.1137/15M1051142.
  47. Andreas Krause and Carlos Guestrin. A note on the budgeted maximization of submodular functions. Technical Report CMU-CALD-05-103, 2005. Google Scholar
  48. Ariel Kulik, Hadas Shachnai, and Tami Tamir. Approximations for monotone and nonmonotone submodular maximization with knapsack constraints. Math. Oper. Res., 38(4):729-739, November 2013. Google Scholar
  49. Oded Lachish. O(log log rank) competitive ratio for the matroid secretary problem. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 326-335, 2014. URL: https://doi.org/10.1109/FOCS.2014.42.
  50. Jon Lee, Vahab S Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM Journal on Discrete Mathematics, 23(4):2053-2078, 2010. Google Scholar
  51. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res., 35(4):795-806, November 2010. Google Scholar
  52. Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. In Proceedings of the 3rd ACM Conference on Electronic Commerce, EC '01, pages 18-28, 2001. Google Scholar
  53. Hui Lin and Jeff Bilmes. Multi-document summarization via budgeted maximization of submodular functions. In Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics, pages 912-920, 2010. Google Scholar
  54. Hui Lin and Jeff Bilmes. A class of submodular functions for document summarization. In Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, pages 510-520, 2011. Google Scholar
  55. Shengzhong Liu, Zhenzhe Zheng, Fan Wu, Shaojie Tang, and Guihai Chen. Context-aware data quality estimation in mobile crowdsensing. In IEEE INFOCOM 2017 - IEEE Conference on Computer Communications, pages 1-9, 2017. URL: https://doi.org/10.1109/INFOCOM.2017.8057033.
  56. Tengyu Ma, Bo Tang, and Yajun Wang. The simulated greedy algorithm for several submodular matroid secretary problems. Theory of Computing Systems, 58(4):681-706, May 2016. URL: https://doi.org/10.1007/s00224-015-9642-4.
  57. Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: An approach based on strongly factor-revealing lps. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, STOC '11, pages 597-606, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993716.
  58. Aranyak Mehta, Amin Saberi, Umesh Vazirani, and Vijay Vazirani. Adwords and generalized online matching. J. ACM, 54(5):22-es, October 2007. URL: https://doi.org/10.1145/1284320.1284321.
  59. Baharan Mirzasoleiman, Stefanie Jegelka, and Andreas Krause. Streaming non-monotone submodular maximization: Personalized video summarization on the fly. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence, AAAI'18/IAAI'18/EAAI'18. AAAI Press, 2018. Google Scholar
  60. G. L. Nemhauser and L. A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of Operations Research, 3(3):177-188, 1978. URL: https://doi.org/10.1287/moor.3.3.177.
  61. G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions-i. Math. Program., 14(1):265-294, December 1978. URL: https://doi.org/10.1007/BF01588971.
  62. Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory, 129(1):192-224, 2006. URL: https://doi.org/10.1016/j.jet.2004.10.007.
  63. Xinghao Pan, Stefanie Jegelka, Joseph E Gonzalez, Joseph K Bradley, and Michael I Jordan. Parallel double greedy submodular maximization. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 27, pages 118-126. Curran Associates, Inc., 2014. Google Scholar
  64. Ramakumar Pasumarthi, Ramasuri Narayanam, and Balaraman Ravindran. Near optimal strategies for targeted marketing in social networks. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, AAMAS '15, pages 1679-1680, Richland, SC, 2015. International Foundation for Autonomous Agents and Multiagent Systems. Google Scholar
  65. Paulo Shakarian, Joseph Salmento, William Pulleyblank, and John Bertetto. Reducing gang violence through network influence based targeting of social programs. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '14, pages 1829-1836, 2014. Google Scholar
  66. Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett., 32(1):41-43, January 2004. Google Scholar
  67. Erik Vee, Sergei Vassilvitskii, and Jayavel Shanmugasundaram. Optimal online assignment with forecasts. In Proceedings of the 11th ACM conference on Electronic commerce, pages 109-118, 2010. Google Scholar
  68. J. Vondrák. Symmetry and approximability of submodular maximization problems. SIAM Journal on Computing, 42(1):265-304, 2013. Google Scholar
  69. Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 67-74, 2008. Google Scholar
  70. Jan Vondrák, Chandra Chekuri, and Rico Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 783-792, 2011. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact