Minimum-cardinality global defensive alliances in general graphs

IF 4.5 3区 管理学 Q1 OPERATIONS RESEARCH & MANAGEMENT SCIENCE
André Rossi, Alok Singh
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Abstract

A subset S of vertices of an undirected graph G is a defensive alliance if at least half of the vertices in the closed neighborhood of each vertex of S are in S. A defensive alliance is a global defensive alliance if it is also a dominating set of G. This paper addresses the problem of finding minimum-cardinality global defensive alliances for general graphs. Two integer linear programming formulations are proposed to address this problem, the second one being an improved version of the first one in which the constraints are attempted for tightening with a cubing-time algorithm. Two new lower bounds on the cardinality of a defensive global alliance are proposed: the first one is based on a linear time algorithm and is shown to be tighter than three of the four lower bounds from the literature, and the second one is derived from the linear programming relaxation of the aforementioned integer linear programming formulations of the problem. An upper bound on the global defensive alliance number is obtained using a greedy peeling algorithm that is shown to be at least as good as an upper bound of the literature, however it is also shown that the proposed algorithm may be unable to find an optimal solution for some graphs. Finally, numerical experiments are carried out on the 78 DIMACS instances and on 75 Erdős-Rényi graphs with up to 10,000 vertices in order to show the effectiveness of the proposed approaches.

Abstract Image

一般图中的最小基数全球防御联盟
无向图G的顶点子集S是一个防御联盟,如果S的每个顶点的封闭邻域中至少有一半的顶点在S中,则防御联盟是一个全局防御联盟,如果它也是G的支配集,则防御联盟是一个全局防御联盟。提出了两个整数线性规划公式来解决这个问题,第二个公式是第一个公式的改进版本,其中尝试使用立方时间算法来收紧约束。提出了防御性全球联盟基数的两个新的下界:第一个下界是基于线性时间算法,并证明比文献中四个下界中的三个更严格;第二个下界是由上述问题的整数线性规划公式的线性规划松弛得到的。使用贪婪剥离算法获得了全局防御联盟数的上界,该算法至少与文献中的上界一样好,但也表明所提出的算法可能无法找到某些图的最优解。最后,在78个DIMACS实例和75个Erdős-Rényi图(多达10,000个顶点)上进行了数值实验,以证明所提出方法的有效性。
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来源期刊
Annals of Operations Research
Annals of Operations Research 管理科学-运筹学与管理科学
CiteScore
7.90
自引率
16.70%
发文量
596
审稿时长
8.4 months
期刊介绍: The Annals of Operations Research publishes peer-reviewed original articles dealing with key aspects of operations research, including theory, practice, and computation. The journal publishes full-length research articles, short notes, expositions and surveys, reports on computational studies, and case studies that present new and innovative practical applications. In addition to regular issues, the journal publishes periodic special volumes that focus on defined fields of operations research, ranging from the highly theoretical to the algorithmic and the applied. These volumes have one or more Guest Editors who are responsible for collecting the papers and overseeing the refereeing process.
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