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 $1-\frac{4}{n}$.