Approximate Maximin Share Allocations in Matroids

The maximin share guarantee is, in the context of allocating indivisible goods to a set of agents, a recent fairness criterion. A solution achieving a constant approximation of this guarantee always exists and can be computed in polynomial time. We extend the problem to the case where the goods collectively received by the agents satisfy a matroidal constraint. Polynomial approximation algorithms for this generalization are provided: a 1/2-approximation for any number of agents, a (1 − ε)-approximation for two agents, and a (8/9 − ε)-approximation for three agents. Apart from the extension to matroids, the (8/9 − ε)approximation for three agents improves on a (7/8 − ε)-approximation by Amanatidis et al. (ICALP 2015).