The problem of Submodular Welfare Maximization (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. 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. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of, and recently Kapralov et al.  showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al.  were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. 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 0.576 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.
|Title of host publication||Integer Programming and Combinatorial Optimization - 20th International Conference, IPCO 2019, Proceedings|
|Editors||Andrea Lodi, Viswanath Nagarajan|
|Number of pages||14|
|State||Published - 2019|
|Event||20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019 - Ann Arbor, United States|
Duration: 22 May 2019 → 24 May 2019
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019|
|Period||22/05/19 → 24/05/19|
Bibliographical noteFunding Information:
Acknowledgment. We thank Nitish Korula, Vahab S. Mirrokni and Morteza Zadi-moghaddam for sharing with us the full version of their paper . The research of Niv Buchbinder was supported by ISF grant 1585/15 and BSF grant 2014414. The research of Moran Feldman and Mohit Garg was supported in part by ISF grant 1357/16. Yuval Filmus is a Taub Fellow—supported by the Taub Foundations. His research was funded by ISF grant 1337/16.
© 2019, Springer Nature Switzerland AG.
- Greedy algorithms
- Online auctions
- Submodular optimization
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science (all)