On the structure of EFX orientations on graphs

TL;DR

This paper investigates the structure of graphs that always admit envy-freeness up to any good (EFX) orientations regardless of valuation, revealing a connection with the graph's chromatic number and providing a full characterization for binary valuations.

Contribution

It introduces the concept of strongly EFX orientable graphs, links this property to the chromatic number, and characterizes such graphs for binary valuations.

Findings

Graphs with chromatic number ≤ 2 are strongly EFX orientable.

Graphs with chromatic number > 3 are not strongly EFX orientable.

Complete characterization of strong EFX orientability for binary valuations.

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 $\chi(G)$ of the graph $G$. In particular, we show that graphs with $\chi(G)\le 2$ are strongly EFX orientable, and those with $\chi(G)>3$ are not strongly EFX orientable. We provide examples of strongly EFX orientable and non-strongly EFX orientable graphs of $\chi(G)=3$ to prove tightness. Finally, we give a complete characterization of strong EFX orientability when restricted to binary valuations.