Approximately EFX and PO Allocations for Bivalued Chores

TL;DR

The paper tackles fair, Pareto-efficient allocations of indivisible chores under bi-valued costs in $\{1,k\}$ with $k>1$. It leverages the Fisher market framework and starts from a $pEF1$ integral equilibrium, performing MPB-feasible reallocations to obtain $(2-1/k)$-EFX and PO for all $\{1,k\}$-instances, improving upon prior $3$-EFX results. Additionally, it provides a specialized polynomial-time algorithm for $k=2$ that achieves EFX and PO, by exploiting stronger structural properties and a refined grouping of agents. The work advances algorithmic guarantees for EFX+PO in chores by exploiting equilibrium-based reallocations while maintaining MPB feasibility, and it highlights open questions about the unconditional existence of EFX for bi-valued chore instances.

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.

Figures (3)

• Figure 1: The illustration of Example \ref{['example:execution_of_reallocation']}. ($a$) The earning status of agents at the beginning of round $t$. ($b$) The earning status of agents after the swap operation.
• Figure 2: The illustration of Example \ref{['example1:execution_of_12reallocation_1']}. ($a$) In this instance, $z = 4$, with $N^z = \{1\}$, $N^{z + 1} = \{2,3,4\}$ and $N^{z + 2} = \{5,6\}$. ($b$) The earning status of agents after reallocation, where $p(X_1) = p(X_5) = 5$, i.e., both of them joins $N^{z+1}$ after the reallocation.
• Figure 3: The illustration of Example \ref{['example1:execution_of_12reallocation_2']}. ($a$) The earning status of agents before reallocation. ($b$) The earning status of agents after reallocation, where agent $1$ receives only high payment items.

Theorems & Definitions (48)

• Definition 2.1: EF
• Definition 2.2: EF1
• Definition 2.3: $\beta$-EFX
• Definition 2.4: PO
• Definition 2.5: Bi-valued Instances
• Lemma 2.6
• proof
• Definition 2.7: pEF1
• Definition 2.8: $\beta$-pEFX
• Lemma 2.9