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The maximum cardinality and maximum weight matching problems can be solved in Õ(m√n) time, a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article, we demonstrate that this “m√n barrier” can be bypassed by approximation. For any ε 0, we give an algorithm that computes a (1 − ε)-approximate maximum weight matching in O(mε−1 log ε−1) time, that is, optimal linear time for any fixed ε. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error.