Stability in Distance Preservation Games on Graphs

TL;DR

This paper introduces graphical distance preservation games on graphs, analyzing the complexity of finding stable agent allocations under various stability notions and parameters.

Contribution

It presents a comprehensive complexity analysis of stability in distance preservation games across different graph topologies, agent counts, and preference structures.

Findings

Complexity varies with graph topology, agent number, and preferences.

Certain stability problems are computationally hard in general.

Parameterization offers insights into tractable cases.

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.