Hybridizing simulated annealing with variable neighborhood search for bipartite graph crossing minimization

Given a bipartite graph, the problem we address, called the bipartite graph crossing minimization problem (BGCMP), is to find an embedding the parts of the graph along two parallel lines so that the number of edge crossings is minimized. It is assumed that each edge is drawn as a straight line segment and edges sharing an end vertex do not cross. We propose an approach for the BGCMP, which combines a simulated annealing (SA) method with a variable neighborhood search (VNS) scheme. These two algorithms are executed iteratively. At each iteration, the solution produced by SA is submitted as input to the VNS component of the approach. Our VNS algorithm uses a local search technique which is based on a fast insertion neighborhood exploration procedure. We show that the time complexity of this procedure is O(n2), where n is the order of the graph. Another fast procedure is proposed for computing the gain in the objective function value obtained by swapping positions of two vertices. We experimentally compare our algorithm (called SA-VNS) against the tabu search algorithm as well as GRASP approach from the literature. Computational results are reported on four sets of bipartite graphs. The results demonstrate the superiority of SA-VNS over the state-of-the-art methods. The source code implementing SA-VNS is made publicly available as a benchmark for future comparisons.