On the structure of EFX orientations on graphs

TL;DR

This work probes strong EFX-orientability in graphical valuations, revealing a deep link to the graph's chromatic number $χ(G)$. It shows that every $χ(G) ≤ 2$ graph is strongly EFX-orientable, while graphs with $χ(G) > 3$ are not, establishing tightness with 3-chromatic examples; it then provides a complete 0-1 valuation characterization via a forest-based neighborhood condition and proves a practical sufficiency result for bipartite (and near-bipartite) graphs. The study also demonstrates a separation between 0-1 and general valuations, giving examples where an EFX orientation exists for 0-1 valuations but not for general monotone valuations, and discusses subdivisions and zero-valued edges as tools to generate forbidden structures. Overall, the results advance understanding of when EFX orientations are guaranteed in graphical settings and identify rich avenues for further exploration, including computational complexity and Pareto-optimality considerations.

Abstract

Fair division is the problem of allocating a set of items among agents in a fair manner. One of the most sought-after fairness notions is envy-freeness (EF), requiring that no agent envies another's allocation. When items are indivisible, it ceases to exist, and envy-freeness up to any good (EFX) emerged as one of its strongest relaxations. The existence of EFX allocations is arguably the biggest open question within fair division. Recently, Christodoulou, Fiat, Koutsoupias, and Sgouritsa (EC 2023) showed that EFX allocations exist for the case of graphical valuations where an instance is represented by a graph: nodes are agents, edges are goods, and each agent values only her incident edges. On the other hand, they showed NP-hardness for checking the existence of EFX orientation where every edge is allocated to one of its incident vertices, and asked for a characterization of graphs that exhibit EFX orientation regardless of the assigned valuations. In this paper, we make significant progress toward answering their question. We introduce the notion of strongly EFX orientable graphs -- graphs that have EFX orientations regardless of how much agents value the edges. We show a surprising connection between this property and the chromatic number $χ(G)$ of the graph $G$. In particular, we show that graphs with $χ(G)\le 2$ are strongly EFX orientable, and those with $χ(G)>3$ are not strongly EFX orientable. We provide examples of strongly EFX orientable and non-strongly EFX orientable graphs of $χ(G)=3$ to prove tightness. Finally, we give a complete characterization of strong EFX orientability when restricted to binary valuations.

Figures (4)

• Figure 1: This figure provides a $K_{2,4}$ graph as an example to demonstrate the characterization of Lemma \ref{['lem:01char']}. If an adversary chooses $T_1$ and $T_2$ as the forest $H$, we can respond by choosing $x_1 \in T_1$ and $x_2 \in T_2$, and $\bigcup_{i=1}^k N_H(x_i)$ would be the red vertices, which is indeed an independent set on $G$. In general, for any forest an adversary chooses on a $K_{2,4}$ (or any bipartite graph), if a tree is just a single vertex its neighbors in the forest form the empty set, so we can choose the vertex and ignore it. Otherwise, we can choose all $x_i$ to be in the same partite set, and its neighbors in the forest will all be from of the same partite set and hence are an independent set on $G$, so this is an example of a graph that fulfills this condition.
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Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4: Zang1998Thomassen2001