research-article

Online Submodular Maximization with Free Disposal

Authors:

T.-H. Hubert Chan,

Shaofeng H.-C. Jiang,

Zhihao Gavin TangAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 4

Article No.: 56, Pages 1 - 29

https://doi.org/10.1145/3242770

Published: 17 September 2018 Publication History

Abstract

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access.

For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α_∞≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased.

Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α_∞≈ 0.3178.

References

[1]

Alexander A. Ageev and Maxim Sviridenko. 2004. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. Comb. Optim. 8, 3 (2004), 307--328.

[2]

B. V. Ashwinkumar and Robert Kleinberg. 2009. Randomized online algorithms for the buyback problem. In Proceedings of the International Workshop on Internet and Network Economics (WINE’09). Springer, 529--536.

Digital Library

[3]

Moshe Babaioff, Jason Hartline, and Robert Kleinberg. 2008. Selling banner ads: Online algorithms with buyback. In Proceedings of the 4th Workshop on Ad Auctions.

[4]

Moshe Babaioff, Jason D. Hartline, and Robert D. Kleinberg. 2009. Selling ad campaigns: Online algorithms with cancellations. In Proceedings of the 10th ACM Conference on Electronic Commerce. ACM, 61--70.

Digital Library

[5]

Siddharth Barman, Seeun Umboh, Shuchi Chawla, and David L. Malec. 2012. Secretary problems with convex costs. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’12) (Lecture Notes in Computer Science), Vol. 7391. Springer, 75--87.

Digital Library

[6]

Mohammad Hossein Bateni, Mohammad Taghi Hajiaghayi, and Morteza Zadimoghaddam. 2013. Submodular secretary problem and extensions. ACM Trans. Algor. 9, 4 (2013), 32. Retrieved from http://dblp.uni-trier.de/db/journals/talg/talg9.html#BateniHZ13.

Digital Library

[7]

Niv Buchbinder and Moran Feldman. 2016. Deterministic algorithms for submodular maximization problems. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’16). SIAM, 392--403.

Digital Library

[8]

Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. 2015. A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM J. Comput. 44, 5 (2015), 1384--1402.

Digital Library

[9]

Niv Buchbinder, Moran Feldman, and Roy Schwartz. 2015. Online submodular maximization with preemption. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15). SIAM, 1202--1216.

Digital Library

[10]

Gruia Calinescu, Chandra Chekuri, Martin Pacal, and Jan Vondracak. 2011. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput. 40, 6 (2011), 1740--1766. arXiv:http://dx.doi.org/10.1137/080733991

Digital Library

[11]

Amit Chakrabarti and Sagar Kale. 2014. Submodular maximization meets streaming: Matchings, matroids, and more. In Integer Programming and Combinatorial Optimization. Springer, 210--221.

Digital Library

[12]

Chandra Chekuri, Shalmoli Gupta, and Kent Quanrud. 2015. Streaming algorithms for submodular function maximization. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’15) (Lecture Notes in Computer Science), Vol. 9134. Springer, 318--330.

[13]

Chandra Chekuri and Amit Kumar. 2004. Maximum coverage problem with group budget constraints and applications. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and the International Workshop on Randomization and Computation (APPROX-RANDOM’04) (Lecture Notes in Computer Science), Vol. 3122. Springer, 72--83.

[14]

Florin Constantin, Jon Feldman, S. Muthukrishnan, and Martin Pál. 2009. An online mechanism for ad slot reservations with cancellations. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 1265--1274.

Digital Library

[15]

Debadeepta Dey, Tian Yu Liu, Martial Hebert, and J. Andrew Bagnell. 2012. Contextual sequence prediction with application to control library optimization. Robotics: Science and Systems VIII. University of Sydney, Sydney, NSW, Australia. http://www.roboticsproceedings.org/rss08/p07.html.

[16]

Shahar Dobzinski and Jan Vondrák. 2012. On the hardness of welfare maximization in combinatorial auctions with submodular valuations. CoRR abs/1202.2792 (2012). Retrieved from http://arxiv.org/abs/1202.2792.

[17]

Uriel Feige. 1998. A threshold of Ln N for approximating set cover. J. ACM 45, 4 (July 1998), 634--652.

Digital Library

[18]

Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. 2011. Maximizing non-monotone submodular functions. SIAM J. Comput. 40, 4 (2011), 1133--1153. arXiv:http://dx.doi.org/10.1137/090779346

Digital Library

[19]

Uriel Feige and Jan Vondrák. 2010. The submodular welfare problem with demand queries. Theory Comput. 6, 1 (2010), 247--290.

[20]

Jon Feldman, Nitish Korula, Vahab S. Mirrokni, S. Muthukrishnan, and Martin Pál. 2009. Online ad assignment with free disposal. In Proceedings of the International Workshop on Internet and Network Economics (WINE’09) (Lecture Notes in Computer Science), Vol. 5929. Springer, 374--385.

Digital Library

[21]

Moran Feldman, Joseph Naor, and Roy Schwartz. 2011. A unified continuous greedy algorithm for submodular maximization. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS’11). IEEE Computer Society, 570--579.

Digital Library

[22]

Moran Feldman, Ola Svensson, and Rico Zenklusen. 2015. A simple O(log log(rank))-competitive algorithm for the matroid secretary problem. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15). SIAM, 1189--1201. Retrieved from http://dl.acm.org/citation.cfm?id=2722129.2722208.

Digital Library

[23]

Moran Feldman and Rico Zenklusen. 2015. The submodular secretary problem goes linear. In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS’15), Venkatesan Guruswami (Ed.). IEEE Computer Society, 486--505.

Digital Library

[24]

Yuval Filmus and Justin Ward. 2012. A tight combinatorial algorithm for submodular maximization subject to a matroid constraint. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE Computer Society, 659--668.

Digital Library

[25]

Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko. 2006. Tight approximation algorithms for maximum general assignment problems. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06). ACM Press, 611--620.

Digital Library

[26]

Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko. 2011. Tight approximation algorithms for maximum separable assignment problems. Math. Oper. Res. 36, 3 (2011), 416--431.

Digital Library

[27]

P. R. Freeman. 1983. The secretary problem and its extensions: A review. Int. Stat. Rev. 51, 2 (1983), 189--206. Retrieved from http://www.jstor.org/stable/1402748.

[28]

Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. 2006. Dependent rounding and its applications to approximation algorithms. J. ACM 53, 3 (2006), 324--360.

Digital Library

[29]

Shayan Oveis Gharan and Jan Vondrák. 2011. Submodular maximization by simulated annealing. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’11), Dana Randall (Ed.). SIAM, 1098--1116.

Digital Library

[30]

Daniel Golovin, Andreas Krause, and Matthew Streeter. 2009. Online learning of assignments that maximize submodular functions. arXiv preprint arXiv:0908.0772 (2009).

[31]

Daniel Golovin, Andreas Krause, and Matthew Streeter. 2014. Online submodular maximization under a matroid constraint with application to learning assignments. arXiv preprint arXiv:1407.1082 (2014).

[32]

Knuth Graham and Donald E. Knuth. 1989. Patashnik, concrete mathematics. In A Foundation for Computer Science.

Digital Library

[33]

Anupam Gupta, Aaron Roth, Grant Schoenebeck, and Kunal Talwar. 2010. Constrained non-monotone submodular maximization: Offline and secretary algorithms. In Proceedings of the International Workshop on Internet and Network Economics (WINE’10) (Lecture Notes in Computer Science), Vol. 6484. Springer, 246--257.

Digital Library

[34]

Xin Han, Yasushi Kawase, and Kazuhisa Makino. 2014. Online unweighted knapsack problem with removal cost. Algorithmica 70, 1 (2014), 76--91.

Digital Library

[35]

Kazuo Iwama and Shiro Taketomi. 2002. Removable online knapsack problems. In Proceedings of the International Colloquium on Automata, Languages, and Programming. Springer, 293--305.

Digital Library

[36]

Frederick Mosteller and John P. Gilbert. 1966. Recognizing the maximum of a sequence. J. Amer. Stat. Assoc. 61, 313 (1966), 35--73. Retrieved from http://www.jstor.org/stable/2283044.

[37]

Samir Khuller, Anna Moss, and Joseph Seffi Naor. 1999. The budgeted maximum coverage problem. Info. Process. Lett. 70, 1 (1999), 39--45.

Digital Library

[38]

Oded Lachish. 2014. O(log log rank) competitive ratio for the matroid secretary problem. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS’14). IEEE Computer Society, 326--335.

Digital Library

[39]

Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. 2009. Non-monotone submodular maximization under matroid and knapsack constraints. In Proceedings of the Symposium on Theory of Computing (STOC’09), Michael Mitzenmacher (Ed.). ACM, 323--332.

Digital Library

[40]

Tengyu Ma, Bo Tang, and Yajun Wang. 2013. The simulated greedy algorithm for several submodular matroid secretary problems. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS’13) (LIPIcs), Vol. 20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 478--489.

[41]

G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. 1978. An analysis of approximations for maximizing submodular set functions—II. Math. Program. 14, 1 (1978), 265--294. Retrieved from

Digital Library

[42]

G. L. Nemhauser and L. A. Wolsey. 1978. Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res. 3, 3 (1978), 177--188. Retrieved from http://www.ams.org/mathscinet-getitem?mr=506656.

Digital Library

[43]

G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. 1978. An analysis of approximations for maximizing submodular set functions—I. Math. Program. 14, 1 (1 Dec. 1978), 265--294.

Digital Library

[44]

Ashwinkumar Badanidiyuru Varadaraja. 2011. Buyback problem-approximate matroid intersection with cancellation costs. In Proceedings of the International Colloquium on Automata, Languages, and Programming. Springer, 379--390.

Digital Library

[45]

Jan Vondrák. 2008. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the Symposium on Theory of Computing (STOC’08). ACM, 67--74.

Digital Library

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Index Terms

Online Submodular Maximization with Free Disposal
1. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Discrete optimization

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 4

October 2018

445 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3266298

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

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Publication History

Published: 17 September 2018

Accepted: 01 July 2018

Revised: 01 April 2018

Received: 01 January 2018

Published in TALG Volume 14, Issue 4

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Liang YZhang FYang KHu Z(2022)A Surface Crack Damage Evaluation Method Based on Kernel Density Estimation for UAV ImagesSustainability10.3390/su14231623814:23(16238)Online publication date: 5-Dec-2022
https://doi.org/10.3390/su142316238
Alaluf NEne AFeldman MNguyen HSuh A(2022)An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality ConstraintMathematics of Operations Research10.1287/moor.2021.122447:4(2667-2690)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1287/moor.2021.1224
Gao RHe ZHuang ZNie ZYuan BZhong Y(2022)Improved Online Correlated Selection2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00123(1265-1276)Online publication date: Feb-2022
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