Crown reductions for the Minimum Weighted Vertex Cover problem

Abstract

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G, w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently 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. Feasible solution:A vertex cover C for G, i.e., a subset C ⊆ V such that for each e ∈ E, e ∩ C = ∅. Goal: To minimize the weight w(C) := u∈C w(u) of the vertex cover C. The unweighted version of the Minimum Vertex Cover problem (shortly, MIN-VC) is the special case of MIN-w-VC with uniform weights w(u) = 1 for each u ∈ V .As the Minimum Vertex Cover problem cannot be solved exactly in polynomial time, unless P = NP, approaches have concentrated on the design of polynomial time approximation algorithms. In spite of a great deal of efforts, the tight bound on its approximability by a polynomial time algorithm is left open. Recall that the problem has a simple 2-approximation algorithm and, for any constant r < 2, no r-approximation algorithm is known, even in the unweighted case. Currently the best lower bound on polynomial time approximability is 10 √ 5 − 21 ≈ 1.36067, due to Dinur and Safra [17]. They proved that achieving a smaller approximation factor is NP-hard.Recently, there has been increasing interest and progress in lowering the exponential running time of algorithms that solve NP-hard optimization problems, like MIN-VC, exactly [11,32]. The theory of parametrized computation and fixed parameter tractability is a newly developed approach dealing with exact algorithms for such intractable problems. Many hard problems can be associated with a parameter in such a way that the problems are tractable when the parameter is fixed or varies within a small range. Such parametrized problems are now known as fixed parameter tractable (FPT) [18]. The parametrized version of the Vertex Cover problem is a well known FPT problem and has received considerable interest:Para...

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