New algorithms for generalized network flows

Abstract

This paper is concerned with generalized network flow problems. In a generalized network, each edge e=(u, v) has a positive “flow multiplier” a _e associated with it. The interpretation is that if a flow of x _e enters the edge at node u, then a flow of a _ex_e exits the edge at v.

The uncapacitated generalized transshipment problem (UGT) is defined on a generalized network where demands and supplies (real numbers) are associated with the vertices and costs (real numbers) are associated with the edges. The goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand, and the sum over the edges of the product of the cost and the flow is minimized. Adler and Cosares [1] reduced the restricted uncapacitated generalized transshipment problem, where only demand nodes are present, to solving a system of linear inequalities with two variables per inequality. The algorithms presented in [2, 3, 4] result in a faster algorithm for restricted UGT.

Generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges (i.e., upper bounds on the amount of flow). The goal is to find a flow such that the flow excesses at the nodes are proportional to the demands and maximized. We present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of UGT. The algorithm can be used to find an optimal flow or an approximation. When used to find a constant factor approximation, the algorithm yields a bound which is not only more efficient than previous algorithms but also strongly polynomial. It is believed to be the first strongly polynomial approximation algorithm for generalized circulation. The existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm.

Research was done while the first author was attending Stanford University and IBM Almaden Research Center. Research partially supported by ONR grant ONR-N00014-91-C-0026 and by NSF PYI Grant CCR-8858097, matching funds from AT&T and DEC.

Research partially supported by ONR-N00014-91-C-0026

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