Efficient Algorithms on Multicommodity Flow over Time Problems with Partial Lane Reversals

Algorithm 2

An FPTAS for the MSMCF problem with partial lane reversals.

 Input: given static multicommodity flow network

 Output: the approximate MSMCF solution with partial lane reversals

 (1) The given network is transformed by adding two-way capacities as .

 (2) Compute the approximate MSMCF on the transformed network by using the algorithm by Fleischer [22].

 (3) Decompose the flow along the paths and cycles and remove flows in cycles .

 (4) Reverse up to the capacity iff , replaced by whenever , where and .

 (5) For each , if is reversed, and . If neither nor is reversed, , where is the saved capacity of .

 (6) Transform the solution to the original network.