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This can be achieved by solving a weighted maximum matching problem, which gives a maximum cardinality matching with maximum sum of weights of edges in the matching.
KARP, An [n.sup.5/2] algorithm for maximum matchings in bipartite graphs, SIAM J.
The determination of the minimum input for a directed network can be converted to a maximum matching problem to solve the network (Figure 2).
If there is no shared head node or tail node on all edges of the network, the network achieves maximum matching. If the network does not match exactly, the value of the driver node [N.sub.D] is equal to the number of nonmatching nodes.
The maximum matching edge of this small network is shown as the red matching edge in the binary graph (Figure 4(c)).
Principle 5 Positive maximum matching first, applying to the situation that appears several reasonable segmentations.
The maximum matching in an undirected graph is a maximum set of edges without common nodes.
Relatively speaking, for the maximum matching of digraphs only a little research has been conducted.
So, the whole algorithm constructs H by first trying all possible subsets of C in time O([2.sup.k]), then trying all possible ways to make H connected by selecting at most O([2.sup.k]) possible sets of types, and then deciding which vertices to use from these types based on a maximum matching algorithm in time O([absolute value of V] + [square root of [absolute value of V]] [absolute value of E])--if the matching does not exist, we skip to the next set of types.
This may be done by finding a maximum matching between the old S and the precolored colors: a precolored color a is adjacent to an old S [member of] P iff no vertex in s is adjacent to a vertex precolored with a in G'.
In subsequent sections, we show how these ideas may be applied to discrete optimisation problems at different levels of complexity: maximum matching ([section]3), graph colouring ([section]3), and Kolmogorov complexity ([section]4).
The problem of finding a maximum matching in a graph G is well known to be solvable in polynomial time (see [22]).

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