Large Cliques Elude the Metropolis Process

In a random graph on n vertices, the maximum clique is likely to be of size very close to 2 lg n. However, the clique produced by applying the naive “greedy” heuristic to a random graph is unlikely to have size much exceeding lg n. The factor of two separating these estimates motivates the search for more effective heuristics. This article analyzes a heuristic search strategy, the Metropolis process, which is just one step above the greedy one in its level of sophistication. It is shown that the Metropolis process takes super-polynomial time to locate a clique that is only slightly bigger than that produced by the greedy heuristic.

[1]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[2]  B. Bollobás,et al.  Cliques in random graphs , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  B. Pittel.  On the probable behaviour of some algorithms for finding the stability number of a graph , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[5]  Colin McDiarmid,et al.  Topics in Chromatic Graph Theory: Colouring random graphs , 2015 .

[6]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  Frank Thomson Leighton,et al.  Graph bisection algorithms with good average case behavior , 1984, Comb..

[8]  Jonathan S. Turner,et al.  Almost All k-Colorable Graphs are Easy to Color , 1988, J. Algorithms.

[9]  A. Sinclair.  Randomised algorithms for counting and generating combinatorial structures , 1988 .

[10]  Bruce E. Hajek,et al.  The time complexity of maximum matching by simulated annealing , 1988, JACM.

[11]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[12]  Rajeev Motwani,et al.  Expanding graphs and the average-case analysis of algorithms for matchings and related problems , 1989, STOC '89.

[13]  Ludek Kucera.  Graphs with Small Chromatic Numbers are Easy to Color , 1989, Inf. Process. Lett..

[14]  Martin E. Dyer,et al.  A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies , 1989, STOC.

[15]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[16]  Galen H. Sasaki.  The Effect of the Density of States on the Metropolis Algorithm , 1991, Inf. Process. Lett..