A branch-and-cut algorithm for the maximum k-balanced subgraph of a signed graph

International audienceWe are interested in the solution of the maximum k-balanced subgraph problem. Let G = (V, E, s) be a signed graph and k a positive scalar. A signed graph is k-balanced if V can be partitioned into at most k sets in such a way that positive edges are found only within the sets and negative edges go between sets. The maximum k-balanced subgraph problem is the problem of finding a subgraph of G that is k-balanced and maximum according to the number of vertices. This problem has applications in clustering problems appearing in collaborative vs conflicting environments. The particular case k = 2 yields the problem of finding a maximum balanced subgraph in a signed graph and its exact solution has been addressed before in the literature. In this paper, we provide a representatives formulation for the general problem and present a partial description of the associated polytope, including the introduction of strengthening families of valid inequalities. A branch-and-cut algorithm is described for finding an optimal solution to the problem. An ILS metaheuristic is implemented for providing primal bounds for this exact method and a branching rule strategy is proposed for the representatives formulation. Computational experiments, carried out over a set of random instances and on a set of instances from an application, show the effectiveness of the valid inequalities and strategies adopted in this work