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

Algorithm 3

Algorithm for MDMCF with partial lane reversals.

Input: given dynamic multicommodity flow network

Output: the MDMCF with partial lane reversals

 (1) A given dynamic network is transformed into time-expanded network by

 (2) An auxiliary network is constructed by adding two-way capacities as .

 (3) Compute the MSMCF on the auxiliary network .

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

 (5) Reverse up to the capacity iff , replaced by 0 whenever where and .

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

 (7) Transform the solution to the original network.