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Online submodular maximization: beating 1/2 made simple

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Abstract

The Submodular Welfare Maximization problem (SWM) captures an important subclass of combinatorial auctions and has been studied extensively. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. For this setting, Korula et al. (SIAM J Comput 47(3):1056–1086, 2018) were able to show that the greedy algorithm is 0.5052-competitive when the items arrive in a uniformly random order. Unfortunately, however, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis of the same algorithm that provides a slightly better guarantee of 0.5096-competitiveness. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.574 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of \(1-1/e\).

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Notes

  1. A set function \(f:2^\mathscr {N}\rightarrow \mathbb {R}\) is monotone if \(f(S) \le f(T)\) for every two sets \(S \subseteq T \subseteq \mathscr {N}\) and submodular if \(f(S \cup \{u\}) - f(S) \ge f(T \cup \{u\}) - f(T)\) for every two such sets and an element \(u \in \mathscr {N}{\setminus } T\).

  2. This constraint on the set of items that can be selected is equivalent to selecting an independent set of the partition matroid \(\mathscr {M}\) defined by the partition \(\{P_1, P_2, \ldots , P_m\}\).

  3. A slightly weaker bound of 19 / 33, with a simpler proof, appeared in the conference version of this paper [4].

  4. A weighted coverage function f is defined as \(f(S) = \sum _{e \in \left( \bigcup _{A \in S} A\right) } w(e)\) for every \(S \subseteq \mathscr {N}\). We remark that such functions are well-known to be monotone and submodular.

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Author information

Author notes

  1. Moran Feldman

    Present address: Department of Computer Science, University of Haifa, 199 Aba Khoushy Ave., Haifa, 3498838, Israel

  2. Mohit Garg

    Present address: IDSIA, USI-SUPSI, Lugano, Switzerland

Authors and Affiliations

  1. Department of Statistics and Operations Research, Tel Aviv University, P.O. Box 39040, 6997801, Tel Aviv, Israel

    Niv Buchbinder

  2. Department of Mathematics and Computer Science, The Open University of Israel, 1 University Road, P.O. Box 808, 4353701, Raanana, Israel

    Moran Feldman & Mohit Garg

  3. Department of Computer Science, Technion — Israel Institute of Technology, Technion City, 3200003, Haifa, Israel

    Yuval Filmus

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  1. Niv Buchbinder
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  2. Moran Feldman
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Correspondence to Moran Feldman.

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An earlier version of this work appeared in IPCO 2019 [4].

We thank Nitish Korula, Vahab S. Mirrokni and Morteza Zadimoghaddam for sharing with us the full version of their paper [27] before it was officially published. The research of Niv Buchbinder was supported by Israel Science Foundation Grant 2233/19 and United States - Israel Binational Science Foundation Grant 2018352. The research of Moran Feldman and Mohit Garg was supported in part by Israel Science Foundation Grant 1357/16. Yuval Filmus is a Taub Fellow—supported by the Taub Foundations. His research was funded by Israel Science Foundation Grant 1337/16.

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Buchbinder, N., Feldman, M., Filmus, Y. et al. Online submodular maximization: beating 1/2 made simple. Math. Program. 183, 149–169 (2020). https://doi.org/10.1007/s10107-019-01459-z

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