Exact and Approximation Algorithms for PMMS Under Identical Constraints

Fair division of resources is a fundamental problem in many disciplines, including computer science, economy, operations research, etc. In the context of fair allocation of indivisible goods, it is well-known that an allocation satisfying the maximum Nash Social Welfare (Max-NSW) is envy-free up to one good (EF1). In this paper, we further consider the relation between a Max-NSW allocation and two well-adopted fairness properties, i.e., envy-free up to any good (EFX) and pairwise maximin share (PMMS). In particular, we show that a Max-NSW allocation is both EFX and PMMS when agents have identical valuation function. Of independent interests, we also provide an algorithm for computing a PMMS allocation for identical variant. Moreover, we show that a $$\frac{4}{5}$$ -PMMS allocation always exists and can be computed in polynomial time when agents have additive valuations and agree on the ordinal ranking of the goods (although they may disagree on the specific cardinal values).