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Approximate EFX and Exact tEFX Allocations for Indivisible Chores: Improved Algorithms

Mahyar Afshinmehr, Matin Ansaripour, Alireza Danaei, Kurt Mehlhorn

TL;DR

It is demonstrated that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous $(2 + \sqrt{6})$ approximation for additive cost functions.

Abstract

We explore the fair distribution of a set of $m$ indivisible chores among $n$ agents, where each agent's costs are evaluated using a monotone cost function. Our focus lies on two fairness criteria: envy-freeness up to any item (EFX) and a relaxed notion, namely envy-freeness up to the transfer of any item (tEFX). We demonstrate that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous $(2 + \sqrt{6})$ approximation for additive cost functions. This result requires extensive case analysis. Christoforidis et al. (IJCAI'24) independently claim the same approximation for additive cost functions; however, we provide a counter-example to their algorithm. We expand the number of agents to any number to get the same approximation guarantee with the assumption of partially identical ordering (IDO) for the cost functions. Additionally, we establish that a tEFX allocation is achievable for three agents if one has an additive 2-ratio bounded cost function, while the others may have general monotone cost functions. This is an improvement from the prior requirement of two agents with additive 2-ratio bounded cost functions. This allocation can also be extended to agent groups with identical valuations. Further, we show various analyses of EFX allocations for chores, such as the relaxations for additive $α$-ratio-bounded cost functions.

Approximate EFX and Exact tEFX Allocations for Indivisible Chores: Improved Algorithms

TL;DR

It is demonstrated that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous approximation for additive cost functions.

Abstract

We explore the fair distribution of a set of indivisible chores among agents, where each agent's costs are evaluated using a monotone cost function. Our focus lies on two fairness criteria: envy-freeness up to any item (EFX) and a relaxed notion, namely envy-freeness up to the transfer of any item (tEFX). We demonstrate that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous approximation for additive cost functions. This result requires extensive case analysis. Christoforidis et al. (IJCAI'24) independently claim the same approximation for additive cost functions; however, we provide a counter-example to their algorithm. We expand the number of agents to any number to get the same approximation guarantee with the assumption of partially identical ordering (IDO) for the cost functions. Additionally, we establish that a tEFX allocation is achievable for three agents if one has an additive 2-ratio bounded cost function, while the others may have general monotone cost functions. This is an improvement from the prior requirement of two agents with additive 2-ratio bounded cost functions. This allocation can also be extended to agent groups with identical valuations. Further, we show various analyses of EFX allocations for chores, such as the relaxations for additive -ratio-bounded cost functions.

Paper Structure

This paper contains 37 sections, 23 theorems, 118 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. EFX and Approximate EFX for Goods
  4. EFX and Approximate EFX for Chores
  5. Our Contribution
  6. Approximate EFX Allocation.
  7. tEFX Allocations.
  8. Preliminaries
  9. Approximate EFX$(\alpha$-EFX)
  10. 2-EFX existence for three agents and $(n-1)$-partial-IDO Cost Functions
  11. 2-EFX Allocation for Three Agents with Subadditive Cost Functions
  12. Constructing Partial Allocation Satisfying Properties (1) and (2) or a Full 2-EFX Allocation
  13. Case A, $\vert \{ c^1_1,c^1_2,c^1_3 \} \vert= 1$:
  14. Case A1, $\vert \{ c^2_1,c^2_2,c^2_3 \} \vert = 1$:
  15. Case A2, $\vert \{ c^{2}_1, c^{2}_2, c^{2}_3 \} \vert = 2$:
  16. ...and 22 more sections

Key Result

lemma 3.1

Let $X$ be an $\alpha$-EFX allocation. After the cycle elimination step (the inner while loop of Algorithm TTECE) on the top trading envy graph, the resulting allocation remains $\alpha$-EFX. Additionally, the procedure runs in polynomial time.

Figures (1)

  • Figure 1: (a) shows the cost matrix: $C_i(c_j)$ is shown in column $C_i$ and row $c_j$. $m_1$ and $m_2$ are any reals such that $\frac{m_1}{2} > m_2 > 6$. (b) shows the initial allocation $X_1 = \{ c_2 \}$, $X_2 = \{ c_3 \}$ and $X_3 = \{ c_1 \}$. $C_i(X_j)$ is shown in column $i$ and row $j$. The corresponding top trading envy graph is shown at the bottom. (c) shows the situation after allocating $c_4$ to agent 2, i.e., it shows the cost of each agent for each bundle and the corresponding top trading envy graph for the partial allocation $X_1 = \{c_2\}$, $X_2 = \{c_3, c_4\}$ and $X_3 = \{c_1\}$. (d) shows the situation after allocating $c_5$ to agent 2, i.e., it shows the cost of each agent for each bundle and the corresponding top trading envy graph for the partial allocation $X_1 = \{c_2\}$, $X_2 = \{c_3, c_4, c_5\}$ and $X_3 = \{c_1\}$. (e) shows the situation after allocating $c_6$ to agent 3, i.e., it shows the cost of each agent for each bundle and the corresponding top trading envy graph for the final allocation $X_1 = \{c_2\}$, $X_2 = \{c_3, c_4, c_5\}$ and $X_3 = \{c_1, c_6\}$.

Theorems & Definitions (49)

  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • definition 2.5: Top Trading Envy Graph
  • lemma 3.1
  • proof
  • lemma 3.2
  • proof
  • theorem 3.3
  • ...and 39 more