Online Submodular Welfare Maximization: Computation is not the Bottleneck

Abstract

We revisit the online submodular welfare maximization problem (OSW), a vast generalization of numerous well-studied online allocation problems, including online bipartite matching, display ads and AdWords. A natural greedy OSW algorithm achieves a competitive ratio of $1/2$ [LLN06], and $1-1/e$ under i.i.d. arrivals [KPV13]. Both bounds are tight for polynomial-time algorithms unless NP=RP [KPV13]. The information-theoretic limits for OSW, in contrast, remained open. This prompted researchers to ask for tighter upper bounds [Mehta13] or conjecture the existence of better-competitive (slow) algorithms [Huang23, LTX+23, CHS24, Huang2024, HTW24}. We resolve Mehta's question and refute the subsequent conjectures, proving that the greedy algorithm is in fact tight for both adversarial and i.i.d. arrivals, regardless of computation time. Seemingly contradicting recent $(1-1/e)$-competitive fractional algorithms for OSW [HJP+24,Patton26], we show that this impossibility even holds for fractional algorithms, provided their outputs correspond to marginals that are roundable in hindsight. Conceptually, our proofs can be seen as identifying that a bottleneck for OSW, even under (deterministic) adversarial arrivals, is the inherently random notion of correlation gap [ADSY12].