Approximate Generalized Matching: $f$-Factors and $f$-Edge Covers

In this paper we present linear time approximation schemes for several generalized matching problems on nonbipartite graphs. Our results include $O_ε(m)$-time algorithms for $(1-ε)$-maximum weight $f$-factor and $(1+ε)$-approximate minimum weight $f$-edge cover. As a byproduct, we also obtain direct algorithms for the exact cardinality versions of these problems running in $O(m\sqrt{f(V)})$ time. The technical contributions of this work include an efficient method for maintaining {\em relaxed complementary slackness} in generalized matching problems and approximation-preserving reductions between the $f$-factor and $f$-edge cover problems.