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

Algorithm 4

An FPTAS algorithm for MDMCF with partial lane reversals.

Input: given dynamic multicommodity flow network

Output: the MDMCF with partial lane reversals

 (1) The auxiliary network is transformed to -condensed auxiliary network with ,

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

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

 (4) Reverse up to the capacity iff , replaced by 0 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.