Theory of equal-flows in networks

Abstract

The Maximum-Flow problem is a classical problem in combinatorial optimization and has many practical applications. We introduce a new variant of this well known Maximum-Flow problem, viz., the Maximum-Equal-Flow problem,wherein, for each vertex (other than the source) in the network, the actual flows along the arcs emanating from that vertex are constrained to be equal and integral. Surprisingly, unlike the Maximum-Flow problem that is known to admit a polynomial time solution, we prove that the Maximum-Equal-Flow problem is NP-Hard. Nevertheless, we provide an approximation algorithm for the Maximum-Equal-Flow problem. We develop a new (analogous) theory for Equal-Flows in networks and also illustrate the Maximum-Equal- Flow equivalents of the fundamental results in flow theory.