Neighborhood Stability in Assignments on Graphs

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.

• The paper demonstrates polynomial-time algorithms that guarantee neighborhood stable assignments in cycle and path graphs.
• It provides counterexamples, using balanced complete bipartite graphs, to show that neighborhood stability is not universal under binary preferences.
• The study introduces a sufficient condition based on the DFVS number, linking seat graph leaves and preference graph structure to ensure stability.

Neighborhood Stability in Assignments on Graphs

The paper "Neighborhood Stability in Assignments on Graphs" by Haris Aziz, Grzegorz Lisowski, Mashbat Suzuki, and Jeremy Vollen explores the assignment problem of allocating agents to vertices of a graph such that no adjacent agents benefit from swapping positions, termed as "neighborhood stability". This study assumes agents' utilities are binary, based solely on their preferences over adjacent vertices. The paper demonstrates that even in this restricted setting, neighborhood stable assignments are not universally guaranteed but focuses on special characterizations that ensure their existence.

Key Contributions

1. Existence in Cycles and Paths:
   • Cycles: The paper establishes that neighborhood stable assignments always exist for cycle seat graphs. Using a constructive polynomial-time algorithm, the authors demonstrated that by leveraging a minimal path partition approach, one can always find a neighborhood stable assignment. This result is significant given that extending the neighborhood beyond immediate neighbors to include agents two steps away often results in non-existence of stable assignments even for small graphs.
   • Paths: Using a similar methodological approach as in cycles, the study shows that for path seat graphs too, a neighborhood stable assignment can always be computed in polynomial time. Their algorithm guarantees that no blocking pairs exist at a distance of two, extending stability slightly beyond immediate adjacency without losing feasibility.
2. Non-Existence Results: Through specific counterexamples, the authors show that neighborhood stable assignments are not always guaranteed. They use a balanced complete bipartite graph to demonstrate a general class of instances where no such stable assignment exists. This establishes a clear boundary to the applicability of neighborhood stability.
3. General Sufficient Condition: The paper introduces a sufficient condition based on the directed feedback vertex set (DFVS) number of the preference graph. Specifically, if the number of leaf nodes in the seat graph exceeds the DFVS number of the preference graph, a neighborhood stable assignment is guaranteed. This result applies to broader graph classes and shows that structure in the preference graph can significantly influence the feasibility of stable assignments.

Implications and Future Research Directions

Practical Implications

1. Role Allocation and Seat Arrangement:
   • The findings can be applied in organizational settings where roles or seats need to be allocated among individuals such that inter-positional swaps do not yield higher utility for anyone. Industries such as event planning, organizational design, and collaborative team role assignments stand to benefit.
2. Hedonic Games and Stability Guarantees:
   • The connection to hedonic games where agents form coalitions based on preferences and seek stability offers theoretical foundations for developing algorithms that ensure community structures are stable.

Theoretical Implications

The study reveals that cycles and paths have inherent structural properties that accommodate neighborhood stability, unlike more complex graphs. This poses interesting theoretical questions about other classes of graphs that might exhibit similar or stronger properties. Further exploration into identifying these classes can expand the scope of stable assignments under different utility models.

Future Directions

1. Exploration of Stability in Other Graph Classes: Extending the exploration to other graph structures, such as grids, trees, and even random graphs, can yield insights into the generalizability of the presented algorithms and techniques.
2. Utility Models beyond Binary Preferences:
   • Investigating the existence and computation of neighborhood stable assignments under cardinal preferences or multi-valued utility models can enrich the theoretical understanding and practical relevance of the study's results.
3. Optimized Algorithms:
   • Developing more efficient algorithms that can handle larger graphs or more complex preference structures while maintaining computational feasibility remains an essential direction for future work.

In conclusion, this paper makes substantial contributions to the domain of assignment problems on graphs, particularly in defining and ensuring neighborhood stability. It opens numerous avenues for both practical applications and theoretical advancements, stimulating deeper exploration into the interplay of preferences, assignments, and stability.