We explore neural network and related heuristic methods for the fast approximate solution of the Maximum Clique problem. One of these algorithms, Mean Field Annealing, is implemented on the Connection Machine CM-5 and a fast annealing schedule is experimentally evaluated on random graphs, as well as on several benchmark graphs. The other algorithms, which perform certain randomized local search operations, are evaluated on the same benchmark graphs, and on Sanchis graphs. One of our algorithms adjusts its internal parameters as its computation evolves. On Sanchis graphs, it finds significantly larger cliques than the other algorithms do. Another algorithm, GSD(;), works best overall, but is slower than the others. All our algorithms obtain significantly larger cliques than other simpler heuristics but run slightly slower; they obtain significantly smaller cliques on average than exact algorithms or more sophisticated heuristics but run considerably faster. All our algorithms are simple...

We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.

Traditionally, practical clique algorithms have been compared based on their performance on various random graphs. We propose a new testing methodology which permits testing to be completed in a fraction of the time required by previous methods. In addition, the range of testing can be extended to include problems that could not be attempted in the past because of being too time consuming. We accomplish this by applying the approach that makes dynamic programming a very effective algorithmic technique: we use tabulated estimates for the time required to solve subproblems rather than timing each exactly. Our computational experiments validate this approach. The next step is to use this workbench to develop fast new algorithms for the maximum clique problem. A mixed algorithm is an algorithm which applies different strategies for the subproblems that arise. Using the dynamic programming approach again for timing, we mechanize the process of developing table driven algorithms which apply ...