Computing Efficient Envy-Free Partial Allocations of Indivisible Goods

TL;DR

This paper investigates the computational complexity of finding envy-free allocations of indivisible goods under relaxed efficiency constraints, revealing both tractable cases and NP-hardness in various scenarios.

Contribution

It introduces a framework for partial envy-free allocations with mild efficiency constraints, analyzing their computational complexity across different utility models.

Findings

Polynomial-time solutions for binary utilities

Fixed-parameter tractability in certain cases

NP-hardness in restricted ternary utility scenarios

Abstract

Envy-freeness is one of the most prominent fairness concepts in the allocation of indivisible goods. Even though trivial envy-free allocations always exist, rich literature shows this is not true when one additionally requires some efficiency concept (e.g., completeness, Pareto-efficiency, or social welfare maximization). In fact, in such case even deciding the existence of an efficient envy-free allocation is notoriously computationally hard. In this paper, we explore the limits of efficient computability by relaxing standard efficiency concepts and analyzing how this impacts the computational complexity of the respective problems. Specifically, we allow partial allocations (where not all goods are allocated) and impose only very mild efficiency constraints, such as ensuring each agent receives a bundle with positive utility. Surprisingly, even such seemingly weak efficiency requirements lead to a diverse computational complexity landscape. We identify several polynomial-time solvable or fixed-parameter tractable cases for binary utilities, yet we also find NP-hardness in very restricted scenarios involving ternary utilities.