Online Submodular Maximization: Beating 1/2 Made Simple

TL;DR

This paper improves the analysis of the greedy algorithm for online submodular maximization in random order settings, achieving a better competitive ratio and extending results to more general matroid constraints.

Contribution

It provides a simpler proof of a slightly improved competitive ratio for the greedy algorithm in online submodular maximization and extends the analysis to general matroid constraints.

Findings

Greedy algorithm is 0.5096-competitive in the random arrival model.

The analysis applies to general partition matroids with a competitive ratio upper bound of 0.576.

The competitive ratio of greedy does not match the offline optimal of 1-1/e.

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 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 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 0.5052-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 1-1/e.