Edge-maximal triangulated subgraphs and heuristics for the maximum clique problem

In this paper, we present a polynomial algorithm that finds an edge-maximal triangulated subgraph of an arbitrary graph. Then, we use this algorithm as a heuristic for the maximum (weight) clique problem. Finally, a local search routine is incorporated into our heuristic. Computational results comparing our algorithm with two existing edge-maximal triangulated subgraph algorithms in the literature show that the subgraphs found by our algorithm tend to contain more edges as well as a better clique of the original graph. Computational results comparing our heuristic with other heuristics, including an efficient randomized heuristic, also show the promise of our heuristic. © 1994 by John Wiley & Sons, Inc.

[1]  Egon Balas,et al.  Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1991, SIAM J. Comput..

[2]  D. Rose.  Triangulated graphs and the elimination process , 1970 .

[3]  D. Rose.  A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS , 1972 .

[4]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[5]  Egon Balas.  A fast algorithm for finding an edge-maximal subgraph with a TR-formative coloring , 1986, Discret. Appl. Math..

[6]  G. Dirac.  On rigid circuit graphs , 1961 .

[7]  Jue Xue.  Fast algorithms for vertex packing and related problems , 1991 .

[8]  Panos M. Pardalos,et al.  A branch and bound algorithm for the maximum clique problem , 1992, Comput. Oper. Res..

[9]  Douglas R. Shier,et al.  Maximal chordal subgraphs , 1988, Discret. Appl. Math..

[10]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[11]  P. Pardalos,et al.  An exact algorithm for the maximum clique problem , 1990 .

[12]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[13]  Panos M. Pardalos,et al.  An algorithm for finding a maximum weighted independent set in an arbitrary graph , 1991, Int. J. Comput. Math..

[14]  Egon Balas,et al.  Addendum: Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1992, SIAM J. Comput..