Optimal Rounding of Instantaneous Fractional Flows Over Time

A transshipment problem with demands that exceed network capacity can be solved by sending ow 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 ow? In this paper, we show that this problem can be solved using minfm; logT; log(m U) 1+log(m U) log(mU) g maximum ow computations and one minimum (convex) cost ow computation. Here m is the number of arcs, 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 ow 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 ow computations and k minimum cost ow computations. Our solutions start with a stationary fractional ow, as described in [5], and use rounding to transform this into an integral ow. The rounding procedure takes O(n) time. email: [email protected]. Department of Industrial Engineering and Operations Research, Columbia University. y email: [email protected]. Sloan School of Management, Massachusetts Institute of Technology. Supported by ONR through grant N00014-96-1-0051.