Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints

TL;DR

This paper investigates the computational complexity of finding Pareto-optimal allocations of indivisible goods with connectivity constraints, revealing NP-hardness in many cases and proposing algorithms for specific scenarios.

Contribution

It characterizes the complexity of Pareto-optimal allocations under connectivity constraints and introduces algorithms for special cases with binary valuations.

Findings

NP-hardness for general graph topologies beyond paths and stars

Existence of instances where Pareto-optimal allocations are not envy-free up to one good

A moving-knife algorithm for binary valuations with non-nested interval structure

Abstract

We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Pareto-optimal allocation satisfies envy-freeness up to one good, and that it is NP-hard to decide whether such an allocation exists, even for binary valuations. We also show that, for a path, it is NP-hard to find a Pareto-optimal allocation that satisfies maximin share, but show that a moving-knife algorithm can find such an allocation when agents have binary valuations that have a non-nested interval structure.