TY - JOUR

T1 - Finding all maximally-matchable edges in a bipartite graph

AU - Tassa, Tamir

PY - 2012/3/16

Y1 - 2012/3/16

N2 - We consider the problem of finding all maximally-matchable edges in a bipartite graph G=(V,E), i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this computation in linear time O(n+m) (where n=|V| and m=|E|). Hence, the time complexity of finding all maximally-matchable edges reduces to that of finding a single maximum matching, which is O(n1 2m) (Hopcroft and Karp [12]), or O(( nlogn)12m) for dense graphs with m=Θ( n2) (Alt et al. [2]). This time complexity improves upon that of the best known algorithms for the problem, which is O(nm) (Costa [5] for bipartite graphs, and Carvalho and Cheriyan [6] for general graphs). Other algorithms for solving that problem are randomized algorithms due to Rabin and Vazirani [15] and Cheriyan [3], the runtime of which is O( n2.376). Our algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when m=O( nr) and r<1.876. In addition, our algorithm is elementary, conceptually simple, and easy to implement.

AB - We consider the problem of finding all maximally-matchable edges in a bipartite graph G=(V,E), i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this computation in linear time O(n+m) (where n=|V| and m=|E|). Hence, the time complexity of finding all maximally-matchable edges reduces to that of finding a single maximum matching, which is O(n1 2m) (Hopcroft and Karp [12]), or O(( nlogn)12m) for dense graphs with m=Θ( n2) (Alt et al. [2]). This time complexity improves upon that of the best known algorithms for the problem, which is O(nm) (Costa [5] for bipartite graphs, and Carvalho and Cheriyan [6] for general graphs). Other algorithms for solving that problem are randomized algorithms due to Rabin and Vazirani [15] and Cheriyan [3], the runtime of which is O( n2.376). Our algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when m=O( nr) and r<1.876. In addition, our algorithm is elementary, conceptually simple, and easy to implement.

KW - Bipartite graphs

KW - Maximally-matchable edges

KW - Maximum matchings

KW - Perfect matchings

UR - http://www.scopus.com/inward/record.url?scp=84857057504&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2011.12.071

DO - 10.1016/j.tcs.2011.12.071

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AN - SCOPUS:84857057504

SN - 0304-3975

VL - 423

SP - 50

EP - 58

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -