EFX Orientations of Multigraphs

TL;DR

This paper investigates EFX orientations in bi-valued symmetric multigraphs with self-loops, revealing NP-completeness of deciding their existence for any multiplicity $q \ge 2$, even under bipartite structure and $α > qβ$. The authors introduce the NTOM concept (non-trivial odd multitree) as a structural barrier: heavy components whose heavy edges induce an NTOM can prevent EFX orientations, while graphs without such a heavy component admit a polynomial-time EFX orientation. The hardness result is established via a CircuitSAT reduction with carefully designed gadgets that enforce bipartiteness and NTOM structure, strengthening our understanding of the boundary between tractable and intractable cases. In contrast, when NTOMs are avoided, the paper provides a constructive, polynomial-time method to obtain EFX orientations, highlighting a dichotomy between NTOM-free instances and NTOM-containing instances. The findings illuminate the landscape between EFX orientations and allocations in multigraphs and suggest further exploration of parameter regimes, such as when $α$ and $β$ are closely related or when relative multiplicities are constrained.

Abstract

We study EFX orientations of multigraphs with self-loops. In this setting, vertices represent agents, edges represent goods, and a good provides positive utility to an agent only if it is incident to the agent. We focus on the bi-valued symmetric case in which each edge has equal utility to both incident agents, and edges have one of two possible utilities $α> β\geq 0$. In contrast with the case of simple graphs for which bipartiteness implies the existence of an EFX orientation, we show that deciding whether a symmetric multigraph $G$ of any multiplicity $q \geq 2$ has an EFX orientation is NP-complete even if $G$ is bipartite, $α> qβ$, and $G$ contains a structure called a non-trivial odd multitree (NTOM). Moreover, we show that NTOMs are a problematic structure in the sense that even very simple NTOMs can fail to have EFX orientations, and multigraphs that do not contain NTOMs always have EFX orientations that can be found in polynomial-time.

Figures (3)

• Figure 1: Example of an NTOM with two vertices having no EFX orientations. The solid edge represents a heavy edge. The dashed self-loops marked with * each represents $q \geq 1$ light edges.
• Figure 2: The multigraph $H$. Each solid (resp. dashed) edge represents a heavy (resp. light) edge.
• Figure 3: The gadgets used in Theorem \ref{['thm:main']}. Each solid (resp. dashed) edge represents a heavy (resp. light) edge, except for the dashed edge marked with * in (d) which represent $q$ light edges.

Theorems & Definitions (21)

• Theorem 1
• proof
• Theorem 3
• Lemma 4
• proof
• Theorem 4
• proof
• Theorem 5
• Lemma 6
• proof