Approximation Algorithms for Computing Maximin Share Allocations

Abstract

We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of \(n\) agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into \(n\) bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a \(2/3\)-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a \(2/3\)-approximation but runs in polynomial time only for a constant number of agents. We then investigate the intriguing case of \(3\) agents, for which it is already known that exact maximin share allocations do not always exist. We provide a \(6/7\)-approximation algorithm for this case, improving on the currently known ratio of \(3/4\). Finally, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in [5, 14], that maximin share allocations exist almost always.