Dynamic Multicommodity Contraflow Problem with Asymmetric Transit Times

Abstract

A maximum dynamic multicommodity flow problem concerns with the transportation of several different commodities through the specific source-sink path of an underlying capacity network with the objective of maximizing the sum of commodity flows within a given time horizon. Motivated by the uneven road condition of transportation network topology, we introduce the dynamic multicommodity contraflow problem with asymmetric transit times on arcs that increase the outbound lane capacities by reverting the orientation of lanes towards the demand nodes. Moreover, a pseudo-polynomial time algorithm by using a time-expanded graph and an FPTAS by using aΔ-condensed time-expanded network are presented.

Optimal Reconfiguration of Network with Variant Transmission Times on Arcs

“…(ii)Non-symmetric. To model, the scenario of uneven road network topology, authors in [27,28,29,30,31,32,33,34] consider the non-symmetric transit times on anti-parallel arcs and modify the idea of Rebennack et al [26] (cf.Figure 1). The travel time of the auxiliary arc is…”

“…Hence, the complexity is also same. Authors in [28] proposed the multisource single-sink EATCF and lexicographic maximum dynamic partial contraflow problems and provided polynomial-time solution utilizing the method of Nath et al [27]. They also extended this approach to generalized dynamic partial contraflow in [29].…”

“…Authors in [28] proposed the multisource single-sink EATCF and lexicographic maximum dynamic partial contraflow problems and provided polynomial-time solution utilizing the method of Nath et al [27]. They also extended this approach to generalized dynamic partial contraflow in [29]. However, for easy reference, we summarize the currently known complexities for single-commodity flow problems with symmetric and asymmetric transit times on anti-parallel arcs in Table 3.…”

“…The static version of the MCNF problem is solved polynomially, whereas the dynamic version is N P-hard [83]. The authors in [30,84,85,86] introduced MDMCF and QMCF with lane reversals and presented approximation algorithms. Gupta et al [87] introduced contrflow approach in the generalized multi-commodity flow problem and provided the solution.…”

“…By using the scaling approach an FPTAS is presented to solve the MDMCF problem in [84]. Authors in [30] modified Algorithm 3 to the case of non-symmetric transit times on anti-parallel arcs by using relation 2.3, and the same approach is applied to compute the solution of EAMCT in pseudo-polynomial time complexity. 4.2.…”