Approximately EFX and PO Allocations for Bivalued Chores

TL;DR

This paper improves algorithms for allocating indivisible chores with bi-valued costs to achieve approximate EFX and Pareto optimality, advancing theoretical guarantees and providing specific solutions for the case when costs are 1 or 2.

Contribution

It introduces a new algorithm that enhances the approximation ratio for EFX and PO allocations in bi-valued chores and offers a specialized solution for the case when costs are 2.

Findings

Improved approximation ratio to (2-1/k) for EFX and PO allocations.

Algorithm starts from integral EF1 equilibrium and reallocates items.

Specialized algorithm for the case when k=2.

Abstract

We consider the computation for allocations of indivisible chores that are approximately EFX and Pareto optimal (PO). Recently, Garg et al. (2024) show the existence of $3$-EFX and PO allocations for bi-valued instances, where the cost of an item to an agent is either $1$ or $k$ (where $k > 1$) by rounding the (fractional) earning restricted equilibrium. In this work, we improve the approximation ratio to $(2-1/k)$, while preserving the Pareto optimality. Instead of rounding fractional equilibrium, our algorithm starts with the integral EF1 equilibrium for bi-valued chores, introduced by Garg et al. (AAAI 2022) and Wu et al. (EC 2023), and reallocates items until approximate EFX is achieved. We further improve our result for the case when $k=2$ and devise an algorithm that computes EFX and PO allocations.