Abstract
Let G=(V,E) be an undirected graph with vertex set V and edge set E. A clique C of G is a subset of the vertices of V with every pair of vertices of C adjacent. A maximum clique is a clique with the maximum number of vertices. A tabu search algorithm for the maximum clique problem that uses an exact algorithm on subproblems is presented. The exact algorithm uses a graph coloring upper bound for pruning, and the best such algorithm to use in this context is considered. The final tabu search algorithm successfully finds the optimal or best known solution for all standard benchmarks considered. It is compared with a state-of-the-art algorithm that does not use exact search. It is slower to find the known optimal solution for most instances but is faster for five instances and finds a larger clique for two instances.
Highlights
Exact algorithms for the maximum clique problem are remarkably efficient, but, for larger problems, heuristic algorithms are necessary
This paper evaluates the option of including an exact algorithm within the tabu search, and considers the best exact algorithm to use
The replacement of the pseudoexact algorithm in HTS with an improved exact algorithm in HTS2 has allowed a pure comparison of tabu search using exact search with a more standard type of tabu search
Summary
Exact algorithms for the maximum clique problem are remarkably efficient, but, for larger problems, heuristic algorithms are necessary. Tabu search is an effective metaheuristic for the maximum clique problem. This paper evaluates the option of including an exact algorithm within the tabu search, and considers the best exact algorithm to use. The candidate exact algorithms all use a graph coloring upper bound. This approach was used in the specific context of permutation code constructions in [1]. The final tabu search algorithm described, modified from a published algorithm [2], finds best known solutions to standard instances but is normally slower than a state-of-the-art tabu search algorithm [3]
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