Neighborhood Stability in Assignments on Graphs

TL;DR

This paper investigates the existence and computation of neighborhood stable assignments in graphs, showing that such assignments always exist for paths and cycles, and providing polynomial algorithms for these cases and other conditions.

Contribution

It introduces the concept of neighborhood stability in agent assignments, proves existence results for specific graph classes, and offers polynomial algorithms for computing stable assignments.

Findings

Neighborhood stable assignments always exist for paths and cycles.

Polynomial-time algorithms are provided for these cases.

A general condition for the existence of neighborhood stable assignments is established.

Abstract

We study the problem of assigning agents to the vertices of a graph such that no pair of neighbors can benefit from swapping assignments -- a property we term neighborhood stability. We further assume that agents' utilities are based solely on their preferences over the assignees of adjacent vertices and that those preferences are binary. Having shown that even this very restricted setting does not guarantee neighborhood stable assignments, we focus on special cases that provide such guarantees. We show that when the graph is a cycle or a path, a neighborhood stable assignment always exists for any preference profile. Furthermore, we give a general condition under which neighborhood stable assignments always exist. For each of these results, we give a polynomial-time algorithm to compute a neighborhood stable assignment.