Abstract
A graph G = (V, E) with its edges labeled in the set {+,-} is called a signed graph. It is balanced if its set of vertices V can be partitioned into two sets V1 and V2, such that all positive edges connect nodes within V1 or V2, and all negative edges connect nodes between V1 and V2. The maximum balanced subgraph problem (MBSP) for a signed graph is the problem of finding a balanced subgraph with the maximum number of vertices. In this work, we present the first polynomial integer linear programming formulation for this problem and a matheuristic to obtain good quality solutions in a short time. The results obtained for different sets of instances show the effectiveness of the matheuristic, optimally solving several instances and finding better results than the exact method in a much shorter computational time.
Highlights
Let G = (V, E) be an undirected graph where V is the set of vertices and E is the set of edges
We developed methods to obtain heuristic solutions for the maximum balanced subgraph problem (MBSP) problem
We provided an alternative formulation for the problem based on the basic definition of a balanced signed graph, whose size grows polynomially as a function of the input size
Summary
The authors stated that a balanced signed graph could be partitioned into two vertex sets, being that all edges within the sets are positive, and all those between the sets are negative. Given a signed graph G = (V, E, s), the MBSP is the problem of finding a subset of vertices V ′ ⊆ V such that the subgraph GV ′ is balanced and maximizes the cardinality of V ′. Branch-and-cut methods presented in [8, 9] use mathematical models based on the fact that a signed graph is balanced if and only if it does not contain a parallel edge or a cycle with an odd number of negative edges [2, 11, 21].
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