Fair Allocation under Conflict Constraints

2026-05-10

AI Summary

This study addresses the fair allocation of indivisible goods under conflict constraints, modeling items as vertices and conflicts as edges in a graph, with each agent required to receive an independent set. Under the dual objectives of EF1 fairness and maximality, the work introduces a color-switching-based local adjustment technique, proving that a maximal EF1 allocation always exists for two agents with monotone valuations and is computable in pseudo-polynomial time. It further establishes the existence of EF[1,1] allocations under non-monotone additive valuations by applying the “cycle plus triangles” theorem to path graphs—a first in this context. Specialized polynomial-time algorithms are developed for interval graphs, bipartite graphs, and paths. However, the paper also shows that for multiple agents or non-monotone valuations, the existence of such allocations is not guaranteed and the corresponding decision problem is NP-hard.

Abstract

We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.

Problem

• fair allocation
• conflict constraints
• indivisible items
• envy-freeness
• maximality

Innovation

• fair allocation
• conflict constraints
• EF1
• color-switching technique
• cycle plus triangles theorem