Throughout this section, let (X, Y, w) be a matching instance.
A matching instance is a triplet (X, Y, w), where X and Y are two sets, and w is a matching cost function for X and Y.
The All- Cavity- MCM problem [ 28] is, given a matching instance (X, Y, w), to compute MCM X, Y ∖ { y}, w) for all y ∈ Y.
We define the All- Pairs- Cavity- MCM problem as, given a matching instance (X, Y, w), to compute MCM(X ∖ { x}, Y ∖ { y}, w) for all x ∈ X and y ∈ Y.
The Min-Cost Bipartite Matching problem (MCM) is, given a matching instance (X, Y, w), to find the minimum cost of a matching between X and Y with respect to w. Denote by MCM X, Y, w) the solution of the MCM problem for the instance (X, Y, w), and call a matching whose cost equals to the solution optimal.
Informally speaking, context reduction is a process that reduces a constraint (pi ) into (Q) if there is a matching instance for (pi ) in (Theta,) that is, there exists ((forall,overline{alpha }.,PRightarrow pi ^{prime })in Theta ) such that (Spi ^{prime }= pi,) for some (S,) and (S,P) reduces to (Q).
Observe that the right-hand side of Equation 4 equals the cost of the bipartite matching M v, v ′ A for the matching instance (N (v ), N (v ′ ), w v, v ′ ) (see Figure 3b).
In all, we get that the total running time required for constructing a flow network N corresponding to the matching instance (X, Y, w) and finding an optimal flow in N is O(n + n m).
If there is no matching instance for (pi ) or no reduction of (S,P) is possible, then (pi ) reduces to itself.
All four dimensions are partially expanded into independent hierarchies where each concept has at least one matching instance in NeuroMorpho.Org.
The process of simplification of a constraint set, also called context reduction, consists of reducing each constraint (pi ) in this set to the context obtained by recursively reducing the context (P) of the matching instance for (pi ) in (Theta ), if such matching exists, until (P=emptyset ) or there exists no instance in (Theta ) that matches with (pi ).
