Minimum vertex blocker clique problem

We study the minimum vertex blocker clique problem (VBCP),¹ which is to remove a subset of vertices of minimum cardinality in a weighted undirected graph, such that the maximum weight of a clique in the remaining graph is bounded above by a given integer r≥1. Cliques are among the most popular concepts used to model cohesive clusters in different graph-based applications, such as social, biological, and communication networks. The general case of VBCP on weighted graphs is known to be NP-hard, and we show that the special case on unweighted graphs is also NP-hard for any fixed integer r≥1. We present an analytical lower bound on the cardinality of an optimal solution to VBCP, as well as formulate VBCP as a linear 0-1 program with an exponential number of constraints. Facet-inducing inequalities for the convex hull of feasible solutions to VBCP are also identified. Furthermore, we develop the first exact algorithm for solving VBCP, which solves the proposed formulation by using a row generation approach. Computational results obtained by utilizing this algorithm on a test-bed of randomly generated instances are also provided.