Algorithms for the fixed linear crossing number problem Original Research Article

Several heuristics and an exact branch-and-bound algorithm are described for the fixed linear crossing number problem (FLCNP). An experimental study comparing the heuristics on a large set of test graphs is given. FLCNP is similar to the 2-page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal “node line” in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In this restricted version of the problem, the order of the vertices along the node line is predetermined and fixed. FLCNP belongs to the class of NP-hard optimization problems (IEEE Trans. Comput. 39 (1) (1990) 124). The heuristics are tested and compared on a variety of graphs including some “real world” instances of interconnection networks proposed as models for parallel computing. The experimental results indicate that a heuristic based on the neural network model yields near-optimal solutions and outperforms the other heuristics. Also, experiments show the exact algorithm to be feasible for graphs with up to 50 edges, in general, although the quality of the initial upper bound is more critical to running time than graph size.