Page Number Problem: an Evaluation of Heuristics and Its Solution Using a Hybrid Evolutionary Algorithm

The page number problem is to determine the minimum number of pages in a book in which a graph G can be embedded with the vertices placed in a sequence along the spine and the edges on the pages of the book such that no two edges cross each other in any drawing. 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 in a single page, (b) statistically evaluated four heuristics for distributing edges on a minimum number of pages with no crossings for a fixed ordering of vertices on the spine and (c) implemented and experimentally evaluated a hybrid evolutionary algorithm (HEA) for solving the pagenumber problem. In accordance with the results of (a) and (b) above, in HEA, placement of vertices on the spine is decided using a random depth first search of the graph and an edge embedding heuristic adapted from Chung et al. is used to distribute the edges on a minimal number of pages. The results of experiments with HEA on selected standard and random graphs show that the algorithm achieves the optimal pagenumber for the standard graphs. HEA performance is also compared with the Genetic Algorithm described by Kapoor et al. It is observed that HEA gives a better solution for most of the graph instances.