Abstract. This paper aims to show that Constraint Programming can be an efficient technique to solve a well-known combinatorial optimization problem: the search for a maximum clique in a graph. A clique of a graph G = (X, E) is a subset V of X, such that every two nodes in V are joined by an edge of E. The maximum clique problem consists of finding ω(G) the largest cardinality of a clique. We propose two new upper bounds of ω(G) and a new strategy to guide the search for an optimal solution. The interest of our approach is emphasized by the results we obtain for the DIMACS Benchmarks. Seven instances are solved for the first time and two better lower bounds for problems remaining open are found. Moreover, we show that the CP method we propose gives good results and quickly.

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A d-dimensional Keller graph has vertices which are numberedwitheachofthe4 d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a “high speed computer the size of a major galaxy”. This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.

A nonlinear least squares formulation for the maximum weight clique problem is proposed. When nonnegativity of variables is relaxed, it becomes possible to enumerate its stationary points assuming the degeneracy did not occur. It is proved that those stationary points are sufficient to recognize certain types of maximal cliques. 1

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A generalization of the Motzkin–Straus theorem

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