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EFX Allocations on Some Multi-graph Classes

Umang Bhaskar, Yeshwant Pandit

TL;DR

The paper extends the existence and computation of EFX allocations from simple graphs to multi-graphs under cancellable valuations, proving polynomial-time results for three broad classes: bipartite multi-graphs, multi-trees, and $t$-chromatic graphs with girth at least $(2t-1)$. It develops iterative, cut-and-choose based algorithms that resolve envy cycles and preserve EFX throughout the process, with a focus on structural graph properties such as bipartiteness, tree structure, and girth-driven color partitions. The results unify and generalize prior work by showing exact EFX allocations in substantial multi-graph families, including cycles with parallel edges, and provide insights into algorithmic fair division on graph-represented instances. The work highlights both the practical algorithmic implications and the remaining open challenges for general multi-graph instances and tighter agent-count regimes.

Abstract

The existence of EFX allocations is one of the most significant open questions in fair division. Recent work by Christodolou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs", EC 2023) establishes the existence of EFX allocations for graphical valuations, when agents are vertices in a graph, items are edges, and each item has zero value for all agents other than those at its end-points. Thus in this setting, each good has non-zero value for at most two agents, and there is at most one good valued by any pair of agents. This marks one of the few cases when an exact and complete EFX allocation is known to exist for arbitrary agents. In this work, we extend these results to multi-graphs, when each pair of vertices can have more than one edge between them. The existence of EFX allocations in multi-graphs is a natural open question given their existence in simple graphs. We show that EFX allocations exist, and can be computed in polynomial time, for agents with cancellable valuations in the following cases: (i) bipartite multi-graphs, (ii) multi-trees with monotone valuations, and (iii) multi-graphs with girth $(2t-1)$, where $t$ is the chromatic number of the multi-graph. The existence in multi-cycles follows from (i) and (iii).

EFX Allocations on Some Multi-graph Classes

TL;DR

The paper extends the existence and computation of EFX allocations from simple graphs to multi-graphs under cancellable valuations, proving polynomial-time results for three broad classes: bipartite multi-graphs, multi-trees, and -chromatic graphs with girth at least . It develops iterative, cut-and-choose based algorithms that resolve envy cycles and preserve EFX throughout the process, with a focus on structural graph properties such as bipartiteness, tree structure, and girth-driven color partitions. The results unify and generalize prior work by showing exact EFX allocations in substantial multi-graph families, including cycles with parallel edges, and provide insights into algorithmic fair division on graph-represented instances. The work highlights both the practical algorithmic implications and the remaining open challenges for general multi-graph instances and tighter agent-count regimes.

Abstract

The existence of EFX allocations is one of the most significant open questions in fair division. Recent work by Christodolou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs", EC 2023) establishes the existence of EFX allocations for graphical valuations, when agents are vertices in a graph, items are edges, and each item has zero value for all agents other than those at its end-points. Thus in this setting, each good has non-zero value for at most two agents, and there is at most one good valued by any pair of agents. This marks one of the few cases when an exact and complete EFX allocation is known to exist for arbitrary agents. In this work, we extend these results to multi-graphs, when each pair of vertices can have more than one edge between them. The existence of EFX allocations in multi-graphs is a natural open question given their existence in simple graphs. We show that EFX allocations exist, and can be computed in polynomial time, for agents with cancellable valuations in the following cases: (i) bipartite multi-graphs, (ii) multi-trees with monotone valuations, and (iii) multi-graphs with girth , where is the chromatic number of the multi-graph. The existence in multi-cycles follows from (i) and (iii).

Paper Structure

This paper contains 10 sections, 8 theorems, 5 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Comparison to work by Afshinmehr et al. AfshinmehrDKMR24
  4. Notation and Preliminaries
  5. multi-graphs.
  6. Fair division.
  7. EFX Allocations in Bipartite Multi-graphs
  8. EFX Allocations for Monotone Valuations in Tree Multi-graphs
  9. EFX Allocations in $t$-Chromatic multi-graphs with Girth at Least $(2t-1)$
  10. Conclusion

Key Result

Proposition 1

Let $v$ be a cancellable function. Then given sets of goods $S_1$, $S_2$, $T_1$, $T_2$ so that set $S_1$ and $S_2$ are disjoint, sets $T_1$, $T_2$ are disjoint, and $v(S_1) \ge v(T_1)$, $v(S_2) \ge v(T_2)$, it follows that $v(S_1 \cup S_2) \ge v(T_1 \cup T_2)$.

Theorems & Definitions (21)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Claim 6
  • proof
  • Claim 7
  • ...and 11 more