Revisiting the Challenges of MaxClique
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The MaxClique problem, finding the maximum complete subgraph in an E-R G(N,p) random graph in the large N limit, is a very early example of a simple problem for which finding any approximate solution within a factor of 2 of the known, probabilistically determined limit, appears to require P=NP. This type of search has practical importance in very large graphs. Ways of viewing the problem in its configuration space are similar to the issues raised in the hard core model as a means of understanding glass formation. Algorithmic approaches run into phase boundaries. And, most appealing, there is an extensive literature of challenges posed for concrete methods of finding maximum naturally occurring as well as artificially hidden cliques. We use the probabilistic approach in a novel way to provide a more insightful test of constructive algorithms for this problem. We show that improvements over existing methods are possible, meeting the challenges for practical problems with N as large as 10^10 and perhaps longer.
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