Online Submodular Maximization: Beating 1/2 Made Simple
Abstract
The Submodular Welfare Maximization problem (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 onebyone and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of 1/2, and recently Kapralov et al. (2013) showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. (2015) were able to show that the same algorithm is 0.5052competitive 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 0.5096competitiveness for the greedy algorithm in the randomarrival 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 11/e.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.05529
 Bibcode:
 2018arXiv180705529B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 68W27