Version 5.0 The LEDA User Manual

that the bipartition (A, B) is given and that all edges are directed from A to B; all of
them return a maximum cardinality matching and a minimum cardinality node cover.
The initial characters of the inventors are used to distinguish between the algorithms.
The common interface is
list MAX_CARD_BIPARTITE_MATCHING_XX(graph& G,
const list& A,
const list& B,
node_array& NC,
bool use_heuristic = true);
where XX is to be replaced by either HK, ABMP, FF, or FFB. All algorithms can be
asked to use a heuristic to find an initial matching. This is the default.
HK stands for the algorithm due to Hopcroft and Karp [44]. It has running time O( √ nm).
ABMP stands for algorithm due to Alt, Blum, Mehlhorn, and Paul [1]. The algorithm has
running time O( √ nm). The algorithm consists of two major phases. In the first phase
all augmenting paths of length less than Lmax are found, and in the second phase the
remaining augmenting paths are determined. The default value of Lmax is 0.1 √ n. Lmax
is an additional optional parameter of the procedure.
FF stands for the algorithm due to Ford and Fulkerson [34]. The algorithm has running
time O(nm) and FFB stands for a simple and slow version of FF. The algorithm FF has
an additional optional parameter use bfs of type bool. If set to true, breadth-first-search
is used in the search for augmenting paths, and if set to false, depth-first-search is used.
Be aware that the algorithms XX change the graph G. They leave the graph structure
unchanged but reorder adjacency lists (and hence change the embedding). If this is
undesirable you must restore the original order of the adjacency lists as follows.
edge_array edge_number(G); int i = 0;
forall_nodes(v,G)
forall_adj_edges(e,G) edge_number[e] = i++;
call matching algorithm;
G.sort_edges(edge_number);
12.7 Bipartite Weighted Matchings and Assignments
( mwb matching )
We give functions