最小权支配集问题的一种有效数学方法
An Efficient Matheuristic for the Minimum-Weight Dominating Set Problem
摘要：
一个图的最小支配集是一个顶点的最小集合，使得图的每个顶点都属于它，或者与这个集合的一个顶点相邻。该数学对象在与社交网络分析、无线网络设计、编码理论和数据采掘等相关的许多应用中具有高度相关性。当顶点权重给定时，最小化支配集的总权重会产生称为最小权重支配集问题的问题变体。为了解决这个问题，我们引入了一个混合数学结合禁忌搜索与整数规划求解器。后者是用来解决子问题，其中只有一小部分的决策变量，选择相对于搜索历史，是免费的，而其他人是固定的。此外，我们引入了自适应惩罚，以促进探索中间不可行的解决方案在搜索过程中，增强算法的扰动和节点消除程序，并利用更丰富的邻域类。大量的实验分析的各种实例类证明了该算法的良好性能，并在搜索成功的每个组件的贡献进行了分析。
英文摘要：
A minimum dominating set in a graph is a minimum set of vertices such that every vertex of the graph either belongs to it, or is adjacent to one vertex of this set. This mathematical object is of high relevance in a number of applications related to social networks analysis, design of wireless networks, coding theory, and data mining, among many others. When vertex weights are given, minimizing the total weight of the dominating set gives rise to a problem variant known as the minimum weight dominating set problem. To solve this problem, we introduce a hybrid matheuristic combining a tabu search with an integer programming solver. The latter is used to solve subproblems in which only a fraction of the decision variables, selected relatively to the search history, are left free while the others are fixed. Moreover, we introduce an adaptive penalty to promote the exploration of intermediate infeasible solutions during the search, enhance the algorithm with perturbations and node elimination procedures, and exploit richer neighborhood classes. Extensive experimental analyses on a variety of instance classes demonstrate the good performance of the algorithm, and the contribution of each component in the success of the search is analyzed.
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