WEIGHTED AND UNWEIGHTED MAXIMUM CLIQUE ALGORITHMS WITH UPPER-BOUNDS FROM FRACTIONAL COLORING

The linear programming relaxation of the minimum vertex coloring probl em, called the fractional coloring problem, is NP-hard. We describe ef ficient approximation procedures for both the weighted and unweighted versions of the problem. These fractional coloring procedures are then used for generating upper bounds for the (weighted or unweighted) max imum clique problem in the framework of a branch-and-bound procedure. Extensive computational testing shows that replacing the standard uppe r bounding procedures based on various integer coloring heuristics wit h the somewhat more expensive fractional coloring procedure results in improvements of the bound by up to one-fourth in the unweighted and u p to one-fifth in the weighted case, accompanied by a decrease in the size of the search tree by a factor of almost two.