Maximum Degree Vertex-Based Algorithm for Maximum Clique Problem

Title & Authors
Maximum Degree Vertex-Based Algorithm for Maximum Clique Problem
Lee, Sang-Un;

Abstract
In this paper, I propose a linear time algorithm devised to produce exact solution to NP-complete maximum clique problem. The proposed algorithm firstly, from a given graph G=(V,E), sets vertex $\small{v_i}$ of the maximum degree $\small{{\Delta}(G)}$ as clique's major vertex. It then selects vertex $\small{v_j}$ of $\small{{\Delta}(G)}$ among vertices $\small{N_G(v_i)}$ that are adjacent to $\small{v_i}$, only to determine $\small{N_G(v_i){\cap}N_G(v_j)}$ as candidate cliques w and $\small{v_k}$. Next it obtains $\small{w=w{\cap}N_G(v_k)}$ by sorting $\small{d_G(v_k)}$ in the descending order. Lastly, the algorithm executes the same procedure on $\small{G{\backslash}w}$ graph to compare newly attained cliques to previously attained cliques so as to choose the lower. With this simple method, multiple independent cliques would also be attainable. When applied to various regular and irregular graphs, the algorithm proposed in this paper has obtained exact solutions to all the given graphs linear time O(n).
Keywords
Maximum clique;Degree;Maximum Degree;Candidate clique;
Language
Korean
Cited by
References
1.
Wikipedia, "Clique (Graph Theory), http://en.wikipedia.org/wiki/Clique_(graph_theory), Wikimedia Foundation Inc., 2014.

2.
Wikipedia, "Karp's 21 NP-complete Problems, http://en.wikipedia.org/wiki/Karp's_21_NP-complete_problems, Wikimedia Foundation Inc., 2014.

3.
C. Umans, "CS21: Decidability and Tractability," Computer Science, California Institute of Technology, http://www.cs.caltech.edu/-umans/cs21/lec21.pdf, 2009.

4.
Q. Ouyang, P. D. Kaplan, S. Liu, and L. Shumao, "DNA Solution of the Maximal Clique Problem," Science Vol. 278, pp. 446-449, 1997.

5.
H. J. Kim. "Finding Clique Using Backtracking Algorithm," University of Washington, 2006.

6.
E. W. Weisstein, "Complete Graph," Mathworld, http://mathworld.wolfram.com/CompleteGraph.html, 2014.

7.
E. W. Weisstein, "Peterson Graph," Mathworld, http://mathworld.wolfram.com/PetersonGraph.html, 2014.

8.

9.
P. R. J. Ostergard, "A Fast Algorithm for the Maximum Clique Problem," Discrete Applied Mathematics, Vol. 120, No. 1-3, pp. 197-207, Aug. 2002.

10.
W. Pullan and H. H. Hoos, "Dynamic Local Search for the Maximum Clique Problem," Journal of Artificial Intelligence Research, Vol. 25, No. 1, pp. 159-185, Jan. 2006.

11.
W. Pullan, "Phased Local Search for the Maximum Clique Problem," Journal of Combinatorial Optimization, Vol. 12, No. 3, pp. 303-323, Nov. 2006.

12.
W. Pullan, F. Mascia, and M. Brunato, "Cooperating Local Search for the Maximum Clique Problem," Journal of Heuristics, Vol. 17, No. 2, pp. 181-199, Apr. 2011.

13.
L. Xiangmei, D. Fei, and Z. Shao, "A Local Search Algorithm for the Maximum Clique Problem," Journal of Convergence Information Technology, Vol. 7, No. 14, pp. 65-70, Aug. 2012.

14.
S. Balaji, "A New Effective Local Search Heuristic for the Maximum Clique Problem," World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical and Quantum Engineering, Vol. 7, No. 5, pp. 523-529, May 2013.

15.
Z. O. Akbari, "A Polynomial-Time Algorithm for the Maximum Clique Problem," IEEE/ACIS 12th International Conference on Computer and Information Science, pp. 503-507, Jun. 2013.

16.
J. Konc and D. Janezic, "An Improved Branch and Bound Algorithm for the Maximum Clique Problem," MATCH Communications in Mathematical and in Computer Chemistry, Vol. 58, No. 3, pp. 569-590, Dec. 2007.

17.
Wikipedia, "Degree (Graph Theory)," http://en.wikipedia.org/wiki/Degree_(graph-th eory), Wikimedia Foundation Inc., 2014.

18.
P. M. Pardalos, "ESI 6448 Discrete Optimization Theory: Cliques, Quasi-cliques and Clique Partitions in Graphs," Department of Industrial and Systems Engineering, University of Florida, 2008.

19.
Y. D. Lyuu, "Clique," Dept. of Computer Science & Information Engineering, National Taiwan University, 2004.

20.
Mukul S. Bansal and V. Ch. Venkaiah, "A Note on Finding a Maximum Clique in a Graph Using BDDs", Australasian Journal of Combinatorics, Vol. 32, No. 1, pp. 253-258, Jan. 2005.

21.