Fair Division Beyond Monotone Valuations with Applications to Equitable Graph Partitioning

TL;DR

This paper explores fair division for agents with non-monotone preferences, establishing existence and algorithms for fair allocations in both divisible and indivisible settings, with applications to equitable graph partitioning.

Contribution

It introduces new fairness guarantees and algorithms for non-monotone valuation classes, extending fair division theory beyond traditional monotone assumptions.

Findings

EF divisions exist for subadditive, nonnegative valuations of cakes.

Existence of EFE3 allocations for indivisible items under subadditive valuations.

Efficient algorithms for equitable partitions with dense subgraphs.

Abstract

This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ). In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EFE3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EFE3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQE3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately dense subgraphs: Given any graph $G=(V, E)$ and integer $k$ (at most $|V|$), there always exists a partition $V_1, V_2, \ldots, V_k$ of the vertex set such that the edge densities within the parts, $V_i$, are additively within four of each other. Further, such a partition can be computed efficiently.