Minimizing the Cost of EFx Allocations

TL;DR

This paper studies the problem of finding envy-free up to any item (EFx) allocations that minimize costs, proving NP-hardness, exploring kernelization, and analyzing approximation limits and special cases.

Contribution

It formally defines the minCost-EFx Allocation problem, proves its NP-hardness, and investigates kernelization, tractability in restricted settings, and approximation hardness.

Findings

NP-hardness with two agents

Existence of polynomial kernels based on item count

Inapproximability results for general costs

Abstract

Ensuring fairness while limiting costs, such as transportation or storage, is an important challenge in resource allocation, yet most work has focused on cost minimization without fairness or fairness without explicit cost considerations. We introduce and formally define the minCost-EFx Allocation problem, where the objective is to compute an allocation that is envy-free up to any item (EFx) and has minimum cost. We investigate the algorithmic complexity of this problem, proving that it is NP-hard already with two agents. On the positive side, we show that the problem admits a polynomial kernel with respect to the number of items, implying that a core source of intractability lies in the number of items. Building on this, we identify parameter-restricted settings that are tractable, including cases with bounded valuations and a constant number of agents, or a limited number of item types under restricted cost models. Finally, we turn to cost approximation, proving that for any $\rho>1$ the problem is not $\rho$-approximable in polynomial time (unless $P=NP$), while also identifying restricted cost models where costs are agent-specific and independent of the actual items received, which admit better approximation guarantees.