APPROXIMATION SCHEMES FOR MAXIMUM CARDINALITY MATCHING BAHMAN KALANTARI and ALI SHOKOUFANDEH
نویسندگان
چکیده
Let G = (V;E) be an undirected graph. Given an odd number k = 2l + 1, a matchingM is said to be k-optimal if it does not admit an augmenting path of length less than or equal to k. We prove jM j jM j(l+1)=(l+2), whereM is a maximum cardinality matching. If M is not already (k+ 2)-optimal, using M , in O(jEj) time we compute a (k+2)-optimal matching. We show that starting with any matching, repeated k-optimizations result in an optimal matching in O( p jM jjEj) time. This approximation scheme extends to the minimum weight perfect matching, and the minimum weight uncapacitated perfect 2-matching problems over complete graphs, and complete bipartite graphs with edge weights of one and two. In particular we obtain a fast approximation algorithm for the traveling salesman problem over complete graphs with edge weights of one and two.
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