Finding nucleolus of flow game

Abstract

We study the algorithmic issues of finding the nucleolus of a flow game. The flow game is a cooperative game defined on a network D=(V,E; ω). The player set is E and the value of a coalition S ⊆ E is defined as the value of a maximum flow from source to sink in the subnetwork induced by S. We show that the nucleolus of the flow game defined on a simple network (ω(e)=1 for each e ∈ E) can be computed in polynomial time by a linear program duality approach, settling a twenty-three years old conjecture by Kalai and Zemel. In contrast, we prove that both the computation and the recognition of the nucleolus are N℘-hard for flow games with general capacity.