Crown reductions for the Minimum Weighted Vertex Cover problem

The paper studies crown reductions for the Minimum Weighted Vertex Cover prob-lem introduced recently for the unweighted case by Fellows et al. ([15], [1]). We show a close relation of crown reductions to Nemhauser and Trotter reductions based on the linear programming relaxation of the problem. So called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some ad-ditional constraints, are also introduced and studied. We show how crown decompositions and strong crown decompositions can be computed in polynomial time. For weighted König-Egervary graphs (G; w) we show how the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be e±ciently computed. Further, for some specific classes of graphs, simple algorithms for the Min-VC problem with a constant approximation factor r < 2 are provided. On the other hand, we conclude that for the regular graphs, or for the Hamiltonian connected graphs, the problem is as hard to approximate as for general graphs. It is demonstrated how the results about strong crown reductions can be used to achieve a linear size problem kernel for some related vertex cover problems.