The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items

TL;DR

This paper investigates the computational complexity and welfare implications of EFX fairness in allocations with few surplus items, revealing sharp dichotomies and bounds depending on the welfare measure and constraints.

Contribution

It establishes complexity dichotomies for EFX with generalized-mean welfare, providing polynomial algorithms for certain cases and hardness results for others, especially with few surplus items.

Findings

NP-hardness for $p eq 0$ in welfare maximization under EFX

Polynomial-time algorithms for $p ightarrow -\infty$ and $p ightarrow 0$

Welfare loss bounds depending on the welfare measure and surplus items

Abstract

Envy-freeness up to any good (EFX) is a central fairness notion for allocating indivisible goods, yet its existence is unresolved in general. In the setting with few surplus items, where the number of goods exceeds the number of agents by a small constant (at most three), EFX allocations are guaranteed to exist, shifting the focus from existence to efficiency and computation. We study how EFX interacts with generalized-mean ($p$-mean) welfare, which subsumes commonly-studied utilitarian ($p=1$), Nash ($p=0$), and egalitarian ($p \rightarrow -\infty$) objectives. We establish sharp complexity dichotomies at $p=0$: for any fixed $p \in (0,1]$, both deciding whether EFX can attain the global $p$-mean optimum and computing an EFX allocation maximizing $p$-mean welfare are NP-hard, even with at most three surplus goods; in contrast, for any fixed $p \leq 0$, we give polynomial-time algorithms that optimize $p$-mean welfare within the space of EFX allocations and efficiently certify when EFX attains the global optimum. We further quantify the welfare loss of enforcing EFX via the price of fairness framework, showing that for $p > 0$, the loss can grow linearly with the number of agents, whereas for $p \leq 0$, it is bounded by a constant depending on the surplus (and for Nash welfare it vanishes asymptotically). Finally we show that requiring Pareto-optimality alongside EFX is NP-hard (and becomes $\Sigma_2^P$-complete for a stronger variant of EFX). Overall, our results delineate when EFX is computationally costly versus structurally aligned with welfare maximization in the setting with few surplus items.