Distributed Selfish Algorithms for the Max-Cut Game

Abstract

The Max-Cut problem consists in splitting in two parts the set of vertices of a given graph so as to maximize the sum of weights of the edges crossing the partition. We here address the problem of computing locally maximum cuts in general undirected graphs in a distributed manner. To achieve this, we interpret these cuts as the pure Nash equilibria of a n-player non-zero sum game, each vertex being an agent trying to maximize her selfish interest. A distributed algorithm can then be viewed as the choice of a policy for every agent, describing how to adapt her strategy to other agents’ decisions during a repeated play. In our setting, the only information available to any vertex is the number of its incident edges that cross, or do not cross the cut. In the general, weighted case, computing such an equilibrium can be shown to be PLS-complete, as it is often the case for potential games. We here focus on the (polynomial) unweighted case, but with the additional restriction that algorithms have to be distributed as described above. First, we describe a simple distributed algorithm for general graphs, and prove that it reaches a locally maximum cut in expected time \(4\Delta |E|\), where \(E\) is the set of edges and \(\Delta \) its maximal degree. We then turn to the case of the complete graph, where we prove that a slight variation of this algorithm reaches a locally maximum cut in expected time \(O(\log \log n)\). We conclude by giving experimental results for general graphs.