Stability in Distance Preservation Games on Graphs

2026-02-17

AI Summary

This study addresses the stability of agent assignments on graph structures, where each agent specifies ideal distances to others, and investigates whether a stable configuration—immune to unilateral deviations—exists. The work introduces the first graph distance-preserving hedonic game model and systematically analyzes three solution concepts: envy-freeness, swap stability, and jump stability. By integrating parameterized complexity theory, graph theory, and combinatorial optimization, it precisely characterizes the computational complexity of the problem across multiple dimensions, including graph topology, number of agents, and preference structure. The paper fully delineates the boundary between polynomial-time solvable and NP-hard cases under various graph classes and preference settings, thereby establishing a comprehensive complexity landscape for this fundamental problem.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a new class of network allocation games called graphical distance preservation games. Here, we are given a graph, called a topology, and a set of agents that need to be allocated to its vertices. Moreover, every agent has an ideal (and possibly different) distance in which to be from some of the other agents. Given an allocation of agents, each one of them suffers a cost that is the sum of the differences from the ideal distance for each agent in their subset. The goal is to decide whether there is a stable allocation of the agents, i.e., no agent would like to deviate from their location. Specifically, we consider three different stability notions: envy-freeness, swap stability, and jump stability. We perform a comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.

Problem

• graphical distance preservation games
• stable allocation
• envy-freeness
• swap stability
• jump stability

Innovation

• graphical distance preservation games
• stability notions
• parameterized complexity
• network allocation games
• ideal distance preferences