In this paper we apply a general variable neighbourhood search (GVNS) to the cyclic bandwidth sum problem (CBSP). In CBSP the vertices of a graph must be laid out in a circle in such a way that the sum of the distances between pairs of vertices connected by an edge is minimized. GVNS uses different neighbourhood operations for its shaking phase and local search phase. Also the initial solution is improved using random variable neighbourhood search. Extensive experiments were carried out on classes of graphs with known results for which optimal values of cyclic bandwidth sum was achieved. On other classes of graphs, values less than known upper bounds were achieved.
Index Terms--Cyclic bandwidth sum minimization problem, graph layout problem, variable neighbourhood search
The human behavior is a combination of several feelings. There are two cases whether a person likes other individual or not. The liking of an individual leads to love and finally happiness. The feelings of love may be in different forms but here considered to be partners' love. There are three aspects of love for the partner: forgetting process (oblivion), the pleasure of being loved (return), and the reaction to the appeal of the partner (instinct). Along with that the appeals and the personalities of the two individuals do not vary in time. This model proves that if the geometric mean reactive-ness to love is smaller than the geometric mean forgetting coefficient and the system is asymptotically stable if the ratio of appeals is greater than the reciprocal of ratio of mutual intensiveness coefficient.
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.
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