The antibandwidth maximization problem (AMP) consists of labeling the vertices of a n-vertex graph G with distinct integers from 1 to n such that the minimum difference of labels of adjacent vertices is maximized. This problem can be formulated as a dual problem to the well known bandwidth problem. Exact results have been proved for some standard graphs like paths, cycles, 2 and 3-dimensional meshes, tori, some special trees etc., however, no algorithm has been proposed for the general graphs. In this paper, we propose a memetic algorithm for the antibandwidth maximization problem, wherein we explore various breadth first search generated level structures of a graph-an imperative feature of our algorithm. We design a new heuristic which exploits these level structures to label the vertices of the graph. The algorithm is able to achieve the exact antibandwidth for the standard graphs as mentioned. Moreover, we conjecture the antibandwidth of some 3-dimensional meshes and complement of power graphs, supported by our experimental results.

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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

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