# An Efficient Branch-and-Bound Algorithm for Finding a Maximum Clique

## Abstract

We present an exact and efficient branch-and-bound algorithm for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular kind of graph. It employs approximate coloring and appropriate sorting of vertices to get an upper bound on the size of a maximum clique. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that our algorithm remarkably outperforms other existing algorithms in general. It has been successfully applied to interesting problems in bioinformatics, image processing, the design of quantum circuits, and the design of DNA and RNA sequences for bio-molecular computation.

## Keywords

Search Space Random Graph Maximum Clique Quantum Circuit Sparse Graph## Preview

Unable to display preview. Download preview PDF.

## References

- 1.E. Balas and C.S. Yu: “Finding a maximum clique in an arbitrary graph,” SIAM J. Comput. 15, pp.1054–1068 (1986).zbMATHCrossRefMathSciNetGoogle Scholar
- 2.D. Bahadur K.C., T. Akutsu, E. Tomita, T. Seki, and A. Fujiyama: “Point matching under non-uniform distortions and protein side chain packing based on efficient maximum clique algorithms,” Genome Informatics 13, pp.143–152 (2002).Google Scholar
- 3.E. Balas, S. Ceria, G. Cornuéjols, and G. Pataki: “Polyhedral methods for the maximum clique problem,” pp.11–28 in [9] (1996).Google Scholar
- 4.I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo: “The Maximum Clique Problem.” In: D.-Z. Du and P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, Supplement vol. A, Kluwer Academic Publishers, pp.1–74 (1999).Google Scholar
- 5.J.-M. Bourjolly, P. Gill, G. Laporte, and H. Mercure: “An exact quadratic 0–1 algorithm for the stable set problem,” pp.53–73 in [9] (1996).Google Scholar
- 6.R. Carraghan and P.M. Pardalos: “An exact algorithm for the maximum clique problem,” Oper. Res. Lett. 9, pp.375–382 (1990).zbMATHCrossRefGoogle Scholar
- 7.T. Fujii and E. Tomita: “On efficient algorithms for finding a maximum clique,” Technical Report of IECE (in Japanese), AL81-113, pp.25–34 (1982).Google Scholar
- 8.K. Hotta, E. Tomita, T. Seki, and H. Takahashi: “Object detection method based on maximum cliques,” Technical Report of IPSJ (in Japanese), 2002-MPS-42, pp.49–56 (2002).Google Scholar
- 9.D. S. Johnson and M. A. Trick, (Eds.): “Cliques, Coloring, and Satisfiability,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol.26, American Mathematical Society (1996).Google Scholar
- 10.S. Kobayashi, T. Kondo, K. Okuda, and E. Tomita: “Extracting globally structure free sequences by local structure freeness,” Technical Report CS 03-01, Dept. of Computer Science, Univ. of Electro-Communications (2003).Google Scholar
- 11.Y. Nakui, T. Nishino, E. Tomita, and T. Nakamura: “On the minimization of the quantum circuit depth based on a maximum clique with maximum vertex weight,” Technical Report of Winter LA Symposium 2002, pp.9.1–9.7 (2003).Google Scholar
- 12.P.R.J. Östergård: “A fast algorithm for the maximum clique problem,” Discrete Appl. Math. 120, pp.197–207 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
- 13.P.M. Pardalos and J. Xue: “The maximum clique problem,” J. Global Optimization 4, pp. 301–328 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
- 14.T. Seki and E. Tomita: “Efficient branch-and-bound algorithms for finding a maximum clique,” Technical Report of IEICE (in Japanese), COMP 2001-50, pp.101–108 (2001).Google Scholar
- 15.E.C. Sewell: “A branch and bound algorithm for the stability number of a sparse graph,” INFORMS J. Comput. 10, pp.438–447 (1998).MathSciNetCrossRefGoogle Scholar
- 16.E. Tomita, Y. Kohata, and H. Takahashi: “A simple algorithm for finding a maximum clique,” Techical Report UEC-TR-C5, Dept. of Communications and Systems Engineering, Univ. of Electro communications (1988).Google Scholar
- 17.D. R. Wood: “An algorithm for finding a maximum clique in a graph,” Oper. Res. Lett. 21, pp.211–217 (1997).zbMATHCrossRefMathSciNetGoogle Scholar