Fair Division with Soft Conflicts

TL;DR

This paper introduces efficient algorithms for fair division of indivisible goods with soft conflicts, achieving near-optimal envy-freeness with minimal conflict violations for multiple agents.

Contribution

It presents a linear-time algorithm for EF1 allocations with bounded conflict violations and a simple round-robin method for identical valuations, advancing fair division with conflicts.

Findings

Linear-time algorithm achieves EF1 with violations close to the worst-case bound.

For identical valuations, a simple round-robin algorithm attains optimal violation bounds.

The methods combine fair division techniques with geometric and cardinality constraints.

Abstract

We study the fair division of indivisible goods with conflicts between pairs of goods, represented by a graph $G = (V, E)$. We consider ``soft'' conflicts: assigning two adjacent goods to the same agent is allowed, but we seek allocations that are envy-free up to one good (EF1) while keeping the number of such conflict violations small. We propose a linear-time algorithm for general additive valuations that finds an EF1 allocation with at most $|E|/n + O(|E|^{1-1/(2n-2)})$ violations, for any constant number of agents $n$. The leading term $|E|/n$ matches the worst-case bound on the number of violations. We use a novel approach that combines an algorithm for fair division with cardinality constraints from Biswas \& Barman (2018) and a geometric ``closest points'' argument. For identical additive valuations, we also propose a simple round-robin-based algorithm that finds an EF1 allocation with at most $|E|/n$ violations.