A branch and bound algorithm for the maximum clique problem

A method to solve the maximum clique problem based on an unconstrained quadratic zero-one programming formulation is presented. A branch and bound algorithm for unconstrained quadratic zero-one programming is given that uses a technique to dynamically select variables for the ordering of the branching tree. Dynamic variable selection is equivalent to vertex selection in a similar branch and bound algorithm for the maximum clique problem. In this paper we compare two different rules for selecting a vertex. The first rule selects a variable corresponding to a vertex with high connectivity (a greedy approach) and the second rule selects a variable corresponding to a vertex with low connectivity (a nongreedy approach). We demonstrate that the first rule discovers a maximum clique sooner but it takes significantly longer to verify optimality. Computational results for an efficient vectorizable implementation on an IBM 3090 are provided for randomly generated graphs with up to 1000 vertices and 150,000 edges.