Computing Welfare-Maximizing Fair Allocations of Indivisible Goods

TL;DR

This paper investigates the computational complexity of finding fair and welfare-maximizing allocations of indivisible goods, revealing NP-hardness results and efficient algorithms for special cases.

Contribution

It provides complexity classifications for fair welfare-maximizing allocations under EF1 and PROP1 fairness notions, including polynomial algorithms for two agents and pseudopolynomial algorithms for fixed agents.

Findings

Both problems are strongly NP-hard with variable agents.

Problem (1) is polynomial-time solvable for two agents.

Pseudopolynomial algorithms are designed for fixed number of agents.

Abstract

We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx).