$$K$$ -page crossing number minimization problem: An evaluation of heuristics and its solution using GESAKP

Abstract

The \(K\)-page crossing number minimization problem (KPMP) is to determine the minimum number of edge crossings over all \(K\)-page book drawings of a graph \(G\) with the vertices placed in a sequence along the spine and the edges on the \(K\)-pages of the book. In this paper we have (a) statistically evaluated five heuristics for ordering vertices on the spine for minimum number of edge crossings with all the edges placed on a single page, (b) statistically evaluated five heuristics for distributing edges on \(K\)-pages with minimum number of crossings for a fixed ordering of vertices on the spine and (c) implemented and experimentally evaluated an instance of guided evolutionary simulated annealing (GESA) called GESAKP here for solving the KPMP. In accordance with the results of (a) and (b) above, in GESAKP, placement of vertices on the spine is decided using a random depth first search of the graph and an edge embedding heuristic is used to distribute the edges on \(K\)-pages of a book. Extensive experiments have been carried out on a suite of benchmark, standard and random graphs to compare the performance of GESAKP with variants of the simple genetic algorithm and other existing approaches. In order to improve the results for some graphs, simple extensions to GESAKP were made. Experiments show that in almost all cases, zero or low cost could be achieved for \(K\le 5\). Also, for \(K \le \) ‘known upper bound’ i.e. upper bound for minimum number of pages necessary to draw or embed the edges of a graph without crossings, zero crossings were obtained. In general, GESAKP outperformed the other techniques. From our experimental results we also present the conjectures for the \(K\)-page crossing number of some complete tripartite graphs and pagenumber of toroidal meshes and a class of complete bipartite and tripartite graphs.