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

Algorithm 1

MSMCF algorithm with partial lane reversals.

 Input: given static multicommodity flow network

Output: the MSMCF with partial lane reversals

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

 (2) Compute the MSMCF on the transformed network .

 (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.