最大团问题与整数规划模型及其修正、复杂性与实现

IF 2.2 3区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES Symmetry-Basel Pub Date : 2023-10-26 DOI:10.3390/sym15111979
Milos Seda
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引用次数: 0

摘要

最大团问题是最优化及相关图论问题中形式多样的问题,具有广泛的应用。由于它的np完备性(不确定多项式时间),问题出现在更大的实例的可解性。与基于近似或随机启发式方法的传统方法不同,我们将重点放在GAMS(通用代数建模系统)环境中整数规划模型的使用上,该模型基于精确方法和复杂的确定性启发式。我们提出了整数模型的修正,推导了它们的时间复杂度,并展示了它们在GAMS中的直接应用。GAMS可以在几分钟内为具有数百个顶点和数千条边的实例找到最大团问题的最优解。对于非常大的实例,在合理的时间内给出最优的良好近似值。与上述所有算法相比,这种方法的一大优点是,即使GAMS没有在选定的时间限制内找到最知名的解决方案,它也会在计算结束时将其值显示为可到达的边界。
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The Maximum Clique Problem and Integer Programming Models, Their Modifications, Complexity and Implementation
The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the use of integer programming models in the GAMS (General Algebraic Modelling System) environment, which is based on exact methods and sophisticated deterministic heuristics incorporated in it. We propose modifications of integer models, derive their time complexities and show their direct use in GAMS. GAMS makes it possible to find optimal solutions to the maximum clique problem for instances with hundreds of vertices and thousands of edges within minutes at most. For extremely large instances, good approximations of the optimum are given in a reasonable amount of time. A great advantage of this approach over all the mentioned algorithms is that even if GAMS does not find the best known solution within the chosen time limit, it displays its value at the end of the calculation as a reachable bound.
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来源期刊
Symmetry-Basel
Symmetry-Basel MULTIDISCIPLINARY SCIENCES-
CiteScore
5.40
自引率
11.10%
发文量
2276
审稿时长
14.88 days
期刊介绍: Symmetry (ISSN 2073-8994), an international and interdisciplinary scientific journal, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical research in as much detail as possible. There is no restriction on the length of the papers. Full experimental and/or methodical details must be provided, so that results can be reproduced.
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