Network Flows In Optimisation Problems And Their Extensions

Network flow problems are among the most important ones in graph theory. Since there are many well-known polynomial algorithms for solving the classical Maximum Flow Problem, we, instead of summarising them, focus on special formulations and their transformation into the basic one, and because other graph theory problems may be formulated with the help of network flow tools, we show how to formulate the Minimum Steiner Tree Problem using the maximum network flow terminology and derive its mathematical model. Finally, we discuss the Integer Maximal Multicommodity Flow Problem. Since this network flow version belongs to the class of NP-hard combinatorial problems, for large scale instances, it must be solved by approximation or heuristic techniques. We present a stochastic heuristic approach based on a simulated annealing algorithm.