Online submodular welfare maximization: Greedy is optimal

TL;DR

This paper establishes the optimality of the Greedy algorithm for online welfare maximization with coverage valuations, showing it is 1/2-competitive in the worst case and (1-1/e)-competitive in stochastic settings, with additional bounds for budget-additive allocation.

Contribution

It proves the optimal competitive ratios of Greedy in both worst-case and stochastic online welfare maximization scenarios, and provides bounds for budget-additive allocation.

Findings

Greedy is 1/2-competitive for coverage valuations in the worst case.

Greedy achieves (1-1/e)-competitiveness in stochastic settings with diminishing returns.

No online algorithm can surpass a 0.612-competitive ratio for budget-additive allocation.

Abstract

We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless $NP = RP$. Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d.items and valuations satisfying diminishing returns is $(1-1/e)$-competitive, which is optimal even for coverage valuations, unless $NP=RP$. For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem.