Leveraging Symmetric Relations for Approximation Coalition Structure Generation

Abstract

Cooperative game theory studied the Coalition Structure Generation (CSG) problem in a characteristic function form, where each coalition is associated with a value. Given n agents, there are \(2^n-1\) coalitions. Hence, in the CSG problem, given a set of \(2^n-1\) coalitions, each associated with a value, we have to find a maximal valued disjoint set of coalitions with the same union as the whole set. The first best approximation ratio obtainable in \(O(2^n)\) time is \(\frac{2}{n}\) from the optimal [8]. Later, Adams et al. [1] presented an algorithm which is capable of generating a solution with a value of \(\frac{2}{3}\) of the optimal in \(O(\sqrt{n}2.83^n)\) time, a solution with a value \(\frac{1}{2}\) of the optimal in \(O(\sqrt{n}2.59^n)\) time and a \(\frac{1}{8}\) approximation in \(O(2^n)\) time. This paper sheds new light on the CSG problem by exploiting the combinatorics and symmetry and proposes an approximate halfway dynamic programming (HDP) algorithm with time complexity \(O(2^{\frac{3n}{2}})\approx O(2.83^n)\) with an approximation ratio of \(1-\frac{4}{n}\).