Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

TL;DR

This paper develops algorithms for fair and efficient allocations of indivisible goods, achieving Pareto-optimality and envy-freeness notions, with improved computational complexity results for various valuation classes.

Contribution

It introduces pseudo-polynomial and strongly polynomial algorithms for EF1+fPO and EQ1+fPO allocations, extending polynomial-time computability beyond binary and identical valuations.

Findings

EF1+PO allocations can be computed in polynomial time for constant number of agents.

EF1+fPO allocations exist and can be computed efficiently for positive valuations.

The problem of computing EF1+fPO lies in the PLS complexity class.

Abstract

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of $k$-ary instances where $k$ is a constant, i.e., each agent has at most $k$ different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.