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A Complete Landscape of EFX Allocations on Graphs: Goods, Chores and Mixed Manna

Bo Li, Minming Li, Tianze Wei, Zekai Wu, Yu Zhou

TL;DR

This work delivers a complete landscape of envy-free up to any item (EFX) allocations on graphs when items can be goods, chores, or mixed manna under doubly monotone valuations. It establishes polynomial-time existence and computation for many variants (notably $EFX^+$ for goods, $EFX_-$/ $EFX_0$ for chores, and several mixed notions) while proving NP-hardness for the strongest allocation/orientation notions in the general mixed-manna setting. The paper also provides NP-completeness results for orientations and offers efficient algorithms for important simple graph families, including trees, stars, and paths. Overall, it clarifies where fair allocations are attainable efficiently and where inherent computational hardness arises, offering a robust foundation for graph-based fair division in practice and theory.

Abstract

We study envy-free up to any item (EFX) allocations on simple graphs where vertices and edges represent agents and items respectively. An agent (vertex) is only interested in items (edges) that are incident to her and all other items always have zero marginal value to her. Christodoulou et al. [EC, 2023] first proposed this setting and studied the case of goods where every item has non-negative marginal values to every agent. In this work, we significantly generalize this setting and provide a complete set of results by considering the allocation of arbitrary items that can be goods, chores, or mixed manna under doubly monotone valuations with a mild assumption. For goods, we complement the results by Christodoulou et al. [EC, 2023] by considering another weaker notion of EFX in the literature and showing that an orientation -- a special allocation where each edge must be allocated to one of its endpoint agents -- that satisfies the weaker notion always exists and can be computed in polynomial time, contrary to the stronger notion for which an orientation may not exist and determining its existence is NP-complete. For chores, we show that an envy-free allocation always exists, and an EFX orientation may not exist but its existence can be determined in polynomial time. For mixed manna, we consider the four notions of EFX in the literature. We prove that an allocation that satisfies the strongest notion of EFX may not exist and determining its existence is NP-complete, while one that satisfies any of the other three notions always exists and can be computed in polynomial time. We also prove that an orientation that satisfies any of the four notions may not exist and determining its existence is NP-complete.

A Complete Landscape of EFX Allocations on Graphs: Goods, Chores and Mixed Manna

TL;DR

This work delivers a complete landscape of envy-free up to any item (EFX) allocations on graphs when items can be goods, chores, or mixed manna under doubly monotone valuations. It establishes polynomial-time existence and computation for many variants (notably for goods, / for chores, and several mixed notions) while proving NP-hardness for the strongest allocation/orientation notions in the general mixed-manna setting. The paper also provides NP-completeness results for orientations and offers efficient algorithms for important simple graph families, including trees, stars, and paths. Overall, it clarifies where fair allocations are attainable efficiently and where inherent computational hardness arises, offering a robust foundation for graph-based fair division in practice and theory.

Abstract

We study envy-free up to any item (EFX) allocations on simple graphs where vertices and edges represent agents and items respectively. An agent (vertex) is only interested in items (edges) that are incident to her and all other items always have zero marginal value to her. Christodoulou et al. [EC, 2023] first proposed this setting and studied the case of goods where every item has non-negative marginal values to every agent. In this work, we significantly generalize this setting and provide a complete set of results by considering the allocation of arbitrary items that can be goods, chores, or mixed manna under doubly monotone valuations with a mild assumption. For goods, we complement the results by Christodoulou et al. [EC, 2023] by considering another weaker notion of EFX in the literature and showing that an orientation -- a special allocation where each edge must be allocated to one of its endpoint agents -- that satisfies the weaker notion always exists and can be computed in polynomial time, contrary to the stronger notion for which an orientation may not exist and determining its existence is NP-complete. For chores, we show that an envy-free allocation always exists, and an EFX orientation may not exist but its existence can be determined in polynomial time. For mixed manna, we consider the four notions of EFX in the literature. We prove that an allocation that satisfies the strongest notion of EFX may not exist and determining its existence is NP-complete, while one that satisfies any of the other three notions always exists and can be computed in polynomial time. We also prove that an orientation that satisfies any of the four notions may not exist and determining its existence is NP-complete.
Paper Structure (28 sections, 24 theorems, 10 figures, 1 table, 6 algorithms)

This paper contains 28 sections, 24 theorems, 10 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Problem and Results
  3. Subsequent Work
  4. Other Related Work
  5. Constrained Fair Allocation
  6. Mixed Manna
  7. Fair Allocation on Graphs
  8. Roadmap
  9. Preliminaries
  10. Fairness Notions
  11. Goods and Chores Instances
  12. Goods Instances
  13. Chores Instances
  14. EFX Orientations for Mixed Instances
  15. $\text{EFX}^+_-$ Orientations
  16. ...and 13 more sections

Key Result

Corollary 1

For any goods instance, an $\text{EFX}^+$ allocation always exist and can be computed in polynomial time.

Figures (10)

  • Figure 1: A gadget where agent $a_i$ must receive $(a_i, a_1^{\Delta})$ if the orientation is $\text{EFX}^+_-$. Each dashed edge is a chore for both its endpoint agents.
  • Figure 2: The graph constructed from the formula $(x_1 \vee x_2 \vee x_3) \wedge (x_1 \vee x_2 \vee \neg x_3) \wedge (\neg x_1 \vee \neg x_2 \vee \neg x_3) \wedge (\neg x_1 \vee \neg x_2 \vee x_3)$, where each edge has the same value to both its endpoint agents, each bold solid edge has a value of $2$, each non-bold solid edge has a value of $1$ and each dashed edge has a value of $-1$.
  • Figure 3: A simple tree (also a star, a path) for which no orientation is $\text{EFX}^0_0$ or $\text{EFX}^0_-$. The dashed edge is a chore for both its endpoint agents. The solid edge is a good for both its endpoint agents.
  • Figure 4: An example for which no allocation is $\text{EFX}^0_0$. Each bold solid edge is priceless to both its endpoint agents.
  • Figure 5: (a) OR gadget, (b) NOT gadget, (c) WIRE gadget, (d) TRUE terminator gadget. In these graphs, each agent has an additive valuation. Each bold solid edge is priceless to both its endpoint agents (e.g., it has an infinitely large value of $+\infty$), each non-bold solid edge has an infinitely small value of $\epsilon_1 > 0$ to both its endpoint agents, each dashed line also has an infinitely small value of $\epsilon_2$ to both its endpoint agents with $\epsilon_1 >\epsilon_2 > 0$.
  • ...and 5 more figures

Theorems & Definitions (74)

  • Definition 1: $\text{EFX}^0$
  • Definition 2: $\text{EFX}^+$
  • Definition 3: $\text{EFX}_0$
  • Definition 4: $\text{EFX}_-$
  • Definition 5: $\text{EFX}^0_0$
  • Definition 6: $\text{EFX}^0_-$
  • Definition 7: $\text{EFX}^+_0$
  • Definition 8: $\text{EFX}^+_-$
  • Corollary 1
  • Proposition 1
  • ...and 64 more