Optimal Coalition Structures in Cooperative Graph Games

TL;DR

This paper investigates the computational complexity of finding optimal coalition structures in weighted graph games, demonstrating hardness results and proposing approximation algorithms for specific graph classes.

Contribution

It establishes the intractability of the problem even for restricted graph families and introduces approximation algorithms for planar, minor-free, and bounded degree graphs.

Findings

Optimal coalition structure problem is NP-hard for general and planar graphs.

Constant factor approximation algorithms are provided for certain graph classes.

The problem remains computationally challenging despite restrictions.

Abstract

Representation languages for coalitional games are a key research area in algorithmic game theory. There is an inherent tradeoff between how general a language is, allowing it to capture more elaborate games, and how hard it is computationally to optimize and solve such games. One prominent such language is the simple yet expressive Weighted Graph Games (WGGs) representation [14], which maintains knowledge about synergies between agents in the form of an edge weighted graph. We consider the problem of finding the optimal coalition structure in WGGs. The agents in such games are vertices in a graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minor-free and bounded degree graphs.