1112.0790Dec 4, 2011

Scaling algorithms for approximate and exact maximum weight matching

Ran Duan, Seth Pettie, Hsin-Hao Su

Abstract

The maximum cardinality and maximum weight matching problems can be solved in time $\tilde{O}(m\sqrt{n})$ , a bound that has resisted improvement despite decades of research. (Here $m$

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and

n

are the number of edges and vertices.) In this article we demonstrate that this "

m\sqrt{n}

barrier" is extremely fragile, in the following sense. For any

ε>0

, we give an algorithm that computes a

(1-ε)

-approximate maximum weight matching in

O(mε^{-1}\logε^{-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. Our second contribution is a new exact maximum weight matching algorithm for integer-weighted bipartite graphs that runs in time