Hopcroft-Karp’s algorithm for maximum cardinality matching in bipartite graphs

Maximum Cardinality Matching Ran Duan (Postdoc at MPI) In this lecture • Concepts of matchings • Hopcroft-Karp’s algorithm for maximum cardinality matching in bipartite graphs • Edmonds’s algorithm for maximum cardinality matching in general graphs Basic Concepts and Notations • In graph G=(V,E), a matching M: ▫ A set of vertex-disjoint edges ▫ Matched vertices: the vertices associated with an edge in M ▫ Free vertices: unmatched vertices Basic Concepts and Notations • Matching M: ▫ A set of vertex-disjoint

edges ▫ Matched vertices: the vertices associated with an edge in M ▫ Free vertices: unmatched vertices • Perfect Matching M: ▫ No free vertices Basic Concepts and Notations • Matching M: ▫ A set of vertex-disjoint edges ▫ Matched vertices: the vertices associated with an edge in M ▫ Free vertices: unmatched vertices • Perfect Matching M: ▫ No free vertices • Maximum Cardinality Matching (MCM) ▫ Maximize |M| • Maximum Weighted Matching (MWM) ▫ Maximize  w(e) eM  Applications of matching problems • Assignment ▫ Minimizing costs in job assigments • Image Feature Matching ▫ Match image features (lines, points) between two images • Building blocks of other algorithms ▫ 1.5-Approximate travelling salesman problem  Christofides algorithm  Find a minimum spanning tree and a minimum weight perfect matching Augmenting paths • Alternating paths: ▫ The edges alternate between M and EM • Augmenting paths ▫ The alternating paths whose both ends are free vertices ▫ One more non-matching edges than matching edges Alternating Cycles • A cycle in which edges alternate between M and EM • Given two matchings M1 and M2, then the subgraph (V, M1⊕M2) is composed of ▫ isolating vertices ▫ alternating paths...

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