Randomized and Deterministic Maximin-share Approximations for Fractionally Subadditive Valuations

Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track

Authors

Hannaneh Akrami, Kurt Mehlhorn, Masoud Seddighin, Golnoosh Shahkarami

Abstract

We consider the problem of guaranteeing maximin-share ($\MMS$) when allocating a set of indivisible items to a set of agents with fractionally subadditive ($\XOS$) valuations. For $\XOS$ valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than $1/2$ of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees $0.219225$ of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to $3/13 = 0.230769$. We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is $1/4$-$\MMS$ ex-ante and $1/8$-$\MMS$ ex-post for $\XOS$ valuations. Moreover, we prove an upper bound of $3/4$ on the ex-ante guarantee for this class of valuations.