Fair Division with Soft Conflicts

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.

Figures (2)

• Figure 1: An instance with $n = 5$ agents such that any EF1 allocation has at least $|E|/n$ violations. The number in each vertex denotes the valuation of the good.
• Figure 2: An example of choosing $n = 3$ goods. When we choose three goods with profile vector $(0, -1)$, the violation increase is the same regardless of the adversary's choice.

Theorems & Definitions (27)

• Definition 1: EF1 Bud11
• Proposition 1
• proof
• Proposition 2
• proof
• Definition 2
• Lemma 1: CKM+19
• proof
• Lemma 2
• proof