Branch and bound for the cutwidth minimization problem

The cutwidth minimization problem consists of finding a linear layout of a graph so that the maximum linear cut of edges (i.e., the number of edges that cut a line between consecutive vertices) is minimized. This paper starts by reviewing previous exact approaches for special classes of graphs as well as a linear integer formulation for the general problem. We propose a branch and bound algorithm based on different lower bounds on the cutwidth of partial solutions. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the small-sized instances (up to 32 vertices) as well as some of the large-sized instances tested (up to 158 vertices) in less than 30 minutes of CPU time. We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.