Computational Complexity of Hedonic Games on Sparse Graphs

TL;DR

This paper investigates the computational complexity of various solution concepts in additively separable hedonic games on sparse graphs, revealing hardness results and efficient algorithms depending on graph structure.

Contribution

It provides new complexity results for ASHGs on sparse graphs, including NP-hardness proofs and a pseudo fixed parameter algorithm based on treewidth.

Findings

Maximum egalitarian solution is NP-hard on graphs of treewidth 2.

Polynomial-time solution exists for trees.

A pseudo fixed parameter algorithm is proposed based on treewidth.

Abstract

The additively separable hedonic game (ASHG) is a model of coalition formation games on graphs. In this paper, we intensively and extensively investigate the computational complexity of finding several desirable solutions, such as a Nash stable solution, a maximum utilitarian solution, and a maximum egalitarian solution in ASHGs on sparse graphs including bounded-degree graphs, bounded-treewidth graphs, and near-planar graphs. For example, we show that finding a maximum egalitarian solution is weakly NP-hard even on graphs of treewidth 2, whereas it can be solvable in polynomial time on trees. Moreover, we give a pseudo fixed parameter algorithm when parameterized by treewidth.