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...

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Bibliographic Details
Published in:Symmetry (Basel) 2023-11, Vol.15 (11), p.1979
Main Author: Seda, Milos
Format: Article
Language:English
Subjects:
Algorithms
Apexes
Brain cancer
clique
Decision making
Environment models
GAMS
Genetic algorithms
Graph theory
Graphs
Heuristic
Heuristic methods
independent set
Integer programming
Linear programming
Mathematical models
Mathematical programming
NP-complete problem
Optimization
Polynomials
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Summary: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.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym15111979