Neighborhood Stability in Assignments on Graphs
Haris Aziz, Grzegorz Lisowski, Mashbat Suzuki, Jeremy Vollen
TL;DR
We study neighborhood stability in hedonic seat arrangements where $n$ agents are assigned to a graph $G=(V,E)$ with binary approvals, and agent utilities are $u_i(\pi)=\sum_{j\in N_\pi(i)} \mathbb{I}[i\rightarrow j]$. We show non-existence in general via a constructed counterexample on bipartite seat graphs, demonstrating that neighborhood stability cannot be guaranteed for all binary-preference instances. We prove existence and provide algorithms on cycles and paths: on cycles, a neighborhood-stable assignment exists for all binary profiles and can be computed in $O(n^3)$; on paths, there is a polynomial-time method yielding neighborhood stability with distance-two blocking avoidance. We also present a broad sufficient condition: if the size $\gamma$ of a directed feedback vertex set of $\mathcal{P}$ satisfies $\gamma \le |L|$, where $|L|$ is the number of leaves of $G$, then a neighborhood-stable assignment exists and can be found in polynomial time; in particular, when $\gamma=0$ (i.e., $\mathcal{P}$ is acyclic) the stable assignment is computable in polynomial time.
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.