Optimal Rounding of Instantaneous Fractional Flows Over Time

Abstract

A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done in the minimum number, T, of waves, and at minimum cost, if costs are piece-wise linear convex functions of the flow? In this paper, we show that this problem can be solved using min{m,logT, l+og(mu)-g(U )} maximum flow computations and one minimum (convex) cost flow computation. Here m is the number of arcs, F is the maximum supply or demand, and U is the maximum capacity. When there is only one sink, this problem can be solved in the same asymptotic time as one minimum (convex) cost flow computation. This improves upon the recent algorithm in [5] which solves the quickest transshipment problem (the above mentioned problem without costs) on k terminals using k logT maximum flow computations and k minimum cost flow computations. Our solutions start with a stationary fractional flow, as described in [5], and use rounding to transform this into an integral flow. The rounding procedure takes O(n) time.

Optimal Flow in Dynamic Networks with Nonlinear Cost Functions on Edges

“…Most previous work involving flow costs in dynamic networks considers linear [4,5] or convex [6,7] cost functions with regard to flow value. This implies that the cost of transporting one unit of flow is the same regardless of how many units are transported at a time, or that the cost per unit is rising with the total number of units.…”