A Different Approach to Maximum Clique Search

S. Szabó, Bogdán Zaválnij
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引用次数: 4

Abstract

The way we tackle NP-hard problems in practical setting has experienced a major shift in recent years. Our view has became more sophisticated with the emergence of the parameterized complexity paradigm. We may distinguish subclasses inside the NP-hard complexity class. The complexity of the problems in different subclasses maybe quite different. The overall conservative estimate of the running time is replaced by a more optimistic estimate. In addition the approach of parameterized algorithms is sometimes able to deal with the more complex problems by dividing the problem into harder and a simpler parts. The easier instance at many times reduces to mere preprocessing step leaving us with only the harder part. In this paper we single out the so-called maximum clique problem as a typical representative of the NP-hard complexity class. We propose an algorithm to solve the maximum clique problem motivated by the above ideas. Many of the available maximum clique solvers are descendants or refined versions of the Carraghan–Pardalos algorithm. (Patric \"Osterg\aa rd's cliquer is being as an exception.) The maximum clique problem as a maximization problem can be reduced to a series of k-clique problems as decision problems. Our main observation is that this route offers a number of advantages. The structure of a k-clique decision problem is simpler than the structure of a maximization problem. It affords additional pruning opportunities based on the available value of k. A large scale numerical experiment indicates that in many occasions the combined search space of the k-clique problems is smaller than the search space of the maximization problem. The solver we propose turns out to be rather efficient. In a number of test problems it beats the best available solvers.
最大团搜索的一种不同方法
近年来,我们在实际环境中解决np难题的方式发生了重大转变。随着参数化复杂性范式的出现,我们的观点变得更加复杂。我们可以在NP-hard复杂度类中区分子类。不同子类中问题的复杂性可能大不相同。对运行时间的总体保守估计被更乐观的估计所取代。此外,参数化算法的方法有时能够通过将问题分为较难和较简单的部分来处理较复杂的问题。很多时候,简单的实例简化为简单的预处理步骤,只留下较难的部分。在本文中,我们挑选出所谓的最大团问题作为NP-hard复杂性类的典型代表。基于上述思想,我们提出了一种求解最大团问题的算法。许多可用的最大团解算器都是Carraghan-Pardalos算法的后代或改进版本。(帕特里克·奥斯特拉德的小圈子是个例外。)作为最大化问题的最大团问题可以简化为一系列作为决策问题的k-团问题。我们的主要观察是,这条路线提供了许多优势。k-团决策问题的结构比最大化问题的结构简单。它根据k的可用值提供了额外的剪枝机会。大规模数值实验表明,在许多情况下,k团问题的组合搜索空间小于最大化问题的搜索空间。我们提出的求解器是相当有效的。在许多测试问题中,它击败了最好的可用解算器。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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