New Approximate Graph Matching andMaximum cut ?

For a graph G with e edges and n vertices, and w(E) as a total edge weight, a maximum cardinality matching (MCM) (resp. maximum weighted matching (MWM)) of G is a maximum subset M of edges (resp. a subset M of edges with a maximum edge weight) such that no two edges of M are incident at a common vertex. The best known algorithm for solving the MCM problem in general graphs (resp. the MWM problem in bipartite graphs) requires O(n 5=2) time (resp. O(n(e + n log n)) time). We rst propose an approximate MCM algorithm that runs in O(e + n) sequential time yielding a matching of size at least e n?1. Next, the proposed MCM algorithm is extended to the weighted case running in O(e + n) time, yielding the size of at least w(E) n?1 , when n is even. When n is odd, the lower bound obtained is w(E)?w(Iv) n?2 , where w(Iv) is the weight on edges incident to vertex v which is minimum over considering all vertices. The results improve the bound known before. The proposed algorithms are extremely simple, and the derived lowerbounds are existentially tight. Based on the approximate MCM technique, we nd an O(e + n) time approximation algorithm for the maximum cut (MAXCUT) problem which nds a cut of size at least b en 2(n?1) c for general graphs (when n is even). The problem of approximate MWM has a number of applications, as well as MAXCUT problem, for example in, Vertex Cover and TSP. Both approximate maximum matching and MAXCUT has an important application in VLSI designs that require very fast estimation (or near-optimal solution) in early design stages.

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