Version 5.0 The LEDA User Manual

12.6 Maximum Cardinality Matchings in Bipartite
Graphs ( mcb matching )
A matching in a graph G is a subset M of the edges of G such that no two share an
endpoint. A node cover is a set of nodes NC such that every edge has at least one endpoint
in NC . The maximum cardinality of a matching is at most the minimum cardinality of a
node cover. In bipartite graph, the two quantities are equal.
list MAX CARD BIPARTITE MATCHING(graph& G)
returns a maximum cardinality matching.
Precondition: G must be bipartite.
list MAX CARD BIPARTITE MATCHING(graph& G, node array& NC )
returns a maximum cardinality matching and a minimum cardinality
node cover NC . The node cover has the same cardinality as the
matching and hence proves the optimality of the matching. Precondition:
G must be bipartite.
bool CHECK MCB(const graph& G, const list& M,
const node array& NC )
checks that M is a matching in G, i.e., that at most one edge in M
is incident to any node of G, that NC is a node cover, i.e., for every
edge of G at least one endpoint is in NC and that M and NC have
the same cardinality. The function writes diagnostic output to cerr,
if one of the conditions is violated.
list MAX CARD BIPARTITE MATCHING(graph& G, const list& A,
const list& B)
returns a maximum cardinality matching. Precondition: G must be
bipartite. The bipartition of G is given by A and B. All edges of G
must be directed from A to B.
list MAX CARD BIPARTITE MATCHING(graph& G, const list& A,
const list& B,
node array& NC )
returns a maximum cardinality matching. A minimal node cover is
returned in NC . The node cover has the same cardinality as the
matching and hence proves the maximality of the matching. Precondition:
G must be bipartite. The bipartition of G is given by A and
B. All edges of G must be directed from A to B.
We offer several implementations of bipartite matching algorithms. All of them require