关于反带宽问题计算方法的注记
A note on computational approaches for the antibandwidth problem
摘要：
本文考虑了反带宽问题，也称为对偶带宽问题、分离问题和最大微分着色问题。给定一个带标签的图（即，一个图的顶点的编号），一个节点的反带宽被定义为它的标号与它的所有相邻顶点的标号的最小绝对差。反带宽问题的目标是找到一个最大化反带宽的标签。该问题在一般图中是NP-难的，并在调度，无线电频率分配，讨厌的设施位置和地图着色等不同领域有应用。近年来，在推导问题的理论界和元分析的设计方面做了大量的工作。然而，最佳已知解值与HarwellBoeing矩阵实例的报告上限之间的最优性差距通常非常大（例如，高达577%）。文献中报道的上界是基于涉及简单图特征的理论界，即，大小、阶数和度，以及一个混合整数规划（MIP）模型。我们提出了新的MIP模型的问题，连同有效的不等式，并设计了一个分支和切割算法和迭代求解算法的基础上。这些算法还包括两个起始启发式和一个原始启发式。我们还提出了一个约束规划方法，并计算上界的基础上的稳定数和色数。我们的计算研究表明，所开发的方法允许找到已证明的最佳解决方案，从文献中的8个实例，其中最佳解决方案是未知的，还提供了11个额外的实例，包括改进的解决方案的值为7个实例减少差距，最大的最优性差距现在是46%。
英文摘要：
In this note, we consider the antibandwidth problem, also known as dual bandwidth problem, separation problem and maximum differential coloring problem. Given a labeled graph (i.e., a numbering of the vertices of a graph), the antibandwidth of a node is defined as the minimum absolute difference of its labeling to the labeling of all its adjacent vertices. The goal in the antibandwidth problem is to find a labeling maximizing the antibandwidth. The problem is NP-hard in general graphs and has applications in diverse areas like scheduling, radio frequency assignment, obnoxious facility location and map-coloring. There has been much work on deriving theoretical bounds for the problem and also in the design of metaheuristics in recent years. However, the optimality gaps between the best known solution values and reported upper bounds for the HarwellBoeing Matrix-instances, which are the commonly used benchmark instances for this problem, are often very large (e.g., up to 577%). The upper bounds reported in literature are based on the theoretical bounds involving simple graph characteristics, i.e., size, order and degree, and a mixed-integer programming (MIP) model. We present new MIP models for the problem, together with valid inequalities, and design a branch-and-cut algorithm and an iterative solution algorithm based on them. These algorithms also include two starting heuristics and a primal heuristic. We also present a constraint programming approach, and calculate upper bounds based on the stability number and chromatic number. Our computational study shows that the developed approaches allow to find the proven optimal solution for eight instances from literature, where the optimal solution was unknown and also provide reduced gaps for eleven additional instances, including improved solution values for seven instances, the largest optimality gap is now 46%.
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