Solution of the Maximum $$k$$ k -Balanced Subgraph Problem

A signed graph $$G=V,E,s$$ G = V , E , s is $$k$$ k -balanced if $$V$$ V can be partitioned into at most $$k$$ k sets in such a way that positive edges are found only within the sets and negative edges go between sets. We study the problem of finding a subgraph of $$G$$ G that is $$k$$ k -balanced and maximum according to the number of vertices. This problem has applications in clustering problems appearing in collaborative $$\times $$ × conflicting environments. We describe a 0-1 linear programming formulation for the problem and implement a first version of a branch-and-cut algorithm based on it. GRASP metaheuristics are used to implement the separation routines in the branch-and-cut. We also propose GRASP and ILS-VND procedures to solve heuristically the problem.