Memetic algorithm for the antibandwidth maximization problem

Abstract

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.

A note on computational approaches for the antibandwidth problem

“…The gaps are calculated as 100•(U B L P −z * )/z * , where U B L P is the value of the respective LP-relaxation, and z * is the value of the best known feasible solution for the instance. For the best solution value, we take the results from Table 6 in Lozano et al (2012) (this table is reproduced as Table 2 in the Appendix), which gives a comparison of the state-of-theart heuristics from Bansal and Srivastava (2011), Duarte et al (2011) and Lozano et al (2012), and also the solution values our algorithms obtained (these results are discussed later in this section in detail). For these runs, we directly solve the LP-relaxation of the compact model (F) (and (F lit ), which we also implemented) without any lifting of coefficients or valid inequalities.…”

“…For certain classes of graphs like Hamming graphs (Dobrev et al 2013), hypercubes (Raspaud et al 2009;Wang et al 2009), complete k-ary trees (Calamoneri et al 2009), caterpillars and spiders (Bekos et al 2013(Bekos et al , 2014 there exist tighter bounds and/or exact algorithms. For general graphs, a variety of (meta-)heuristic approaches exist: Bansal and Srivastava (2011) proposed a memetic algorithm, Duarte et al (2011) develops a generalized randomized adaptive search procedure with path relinking, Lozano et al (2012) presented a variable neighborhood search and Scott and Hu (2014) designed a hill-climbing algorithm. Duarte et al (2011) also introduced a mixed-integer programming (MIP) model for the exact solution of the ABP, see Sect.…”

“…The first starting heuristic is in similar spirit to construction heuristics used in e.g., Bansal and Srivastava (2011), Duarte et al (2011) and Lozano et al (2012). Given a vertex i * ∈ V , we construct a breadth-first-search (bfs) tree T i * starting from i * .…”

The Maximum k-Differential Coloring Problem

“…It is worth mentioning though that the maximum differential coloring problem is also known as "dual bandwidth" [8] and "antibandwidth" [5], since it is the complement of the bandwidth minimization problem [9]. Due to the hardness of the problem, heuristics are often used for coloring general graphs, e.g., LP-formulations [10], memetic algorithms [11] and spectral based methods [12]. The differential chromatic number is known only for special graph classes, such as Hamming graphs [13], meshes [14], hypercubes [14,15], complete binary trees [16], complete m-ary trees for odd values of m [5], other special types of trees [16], and complements of interval graphs, threshold graphs and arborescent comparability graphs [17].…”

A note on maximum differential coloring of planar graphs

“…The maximum differential coloring problem is also known as the "anti-bandwidth problem" [3]. Heuristics for the maximum differential coloring problem have been suggested by Duarte et al [5] using LP-formulation, by Bansal et al [2] using memetic algorithms and by Hu et al [12] using spectral based methods. Another line of research focuses on solving the maximum differential coloring problem optimally for special classes of graphs, e.g., Hamming graphs [4], meshes [25], hypercubes [23,26], complete binary trees [27] and complete k-ary trees for odd values of k [3].…”