Approximated vertex cover for graphs with perfect matchings

Chen and Kanj considered the VERTEX COVER problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general VERTEX COVER to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general VERTEX COVER as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.069 + 0.069 (d) over bar where (d) over bar is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly 2 - 6.74/d+6.28. Our algorithm also works for graphs with "large" matchings although its approximation factor is degenerated.