The last two sections address optimization problems modeled on network structures, particularly the shortest path problem and the

maximum flow problem, and discrete optimization problems where the variables are constrained to take integer values.

Next we detail how to transform a maximum weighted triple matching problem to a minimum cost maximum flow problem. The flow network graph [G.sub.f] is constructed based on the triple matching graph [G.sub.3m], as shown in Figure 4.

Therefore the maximum triple matching problem is transformed as a minimum cost maximum flow problem.

Given a directed network flow with unique source s and unique sink t, one of the common problems is the maximum flow problem where the value of flow from s to t should be maximized.

Orlin, "A capacity scaling algorithm for the constrained maximum flow problem," Networks, vol.

Orlin, "A fast and simple algorithm for the maximum flow problem," Operations Research, vol.

MOLP Formulation of minimum cost and

maximum flow problemThe maximum flow problem is logspace complete for P.

Gomory and Hu [35] introduced the concept of a flow equivalent tree and observed that the edge connectivity could be found by solving only n--1 maximum flow problems.

The maximum flow problem and its dual, the minimum cut problem, are classical combinatorial problems with a wide variety of scientific and engineering applications.

The network simplex method of Dantzig [1951] for the transportation problem solves the maximum flow problem as a natural special case.

Schrijver, "On the history of the transportation and

maximum flow problems, " Mathematical Programming, vol.