On approximating minimum vertex cover for graphs with perfect matching

• Published in: Theoretical computer science Vol. 337; no. 1; pp. 305 - 318
• Main Authors: Chen, Jianer, Kanj, Iyad A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 09.06.2005; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation algorithm; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph matching; Graph theory; Inapproximability; Information retrieval. Graph; Mathematics; Sciences and techniques of general use; Theoretical computing; Vertex cover; Vertex cover; Approximation algorithm; Inapproximability; Graph matching; Vertex; Computer theory; Graph degree; Vertex(graph); Perfect matching; Polynomial time
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2004.12.034

It has been a challenging open problem whether there is a polynomial time approximation algorithm for the V ERTEX C OVER problem whose approximation ratio is bounded by a constant less than 2. In this paper, we study the V ERTEX C OVER problem on graphs with perfect matching (shortly, VC-PM). We show that if the VC-PM problem has a polynomial time approximation algorithm with approximation ratio bounded by a constant less than 2, then so does the V ERTEX C OVER problem on general graphs. Approximation algorithms for VC-PM are developed, which induce improvements over previously known algorithms on sparse graphs. For example, for graphs of average degree 5, the approximation ratio of our algorithm is 1.414, compared with the previously best ratio 1.615 by Halldórsson and Radhakrishnan.