Quickest Multi-Commodity Flow Over Time with Partial Lane Reversals

Abstract

Routing of more than one different commodity from specific origin nodes to the corresponding destination nodes through the arcs of an underlying network respecting the capacity constraints is one of the main problems in operational research. Among them, the quickest multicommodity flow problem concerns with minimization of time taken to complete this process. The general multi-commodity and quickest multicommodity flow problems are computationally hard. By flipping the orientation of lanes towards the demand nodes, the outbound lane capacities are increases. We introduce lane reversals in the quickest multicommodity flow problem and present two approximation algorithms, one polynomial-time with the help of length-bounded flow and another FPTAS by using ∆-condensed time-expanded graph. Both algorithms prevent reversing arc capacities that are not required by the optimal flows that may be of interest for other purposes.

Approximate Algorithms for Continuous-Time Quickest Multi-Commodity Contraflow Problem

“…Moreover, our algorithm saves the capacity of lanes that are not necessary to reverse for reducing the time into quickest one. This extends the network flow models introduced in [8] and [12] into partial contraflow framework introduced in [5] and [27]. Proof.…”

“…Similarly, the quickest time with contraflow (cf. Figure (1)(c)) for Commodity-1 and Commodity-2 are T = [5,6) and T = [6, 7) so that the minimum time to satisfy both demands is T = 7. This shows that due to contraflow, time is improved by 12.5 percent.…”

“…Authors in [26] introduced partial contraflow with path reversals. Recently, Dhamala et al [5] presented two approximation algorithms for the QMCF problem with a partial contraflow in the discrete-time setting.…”

Dynamic Multicommodity Contraflow Problem with Asymmetric Transit Times

“…Flow with intermediate storage was investigated by Pyakurel and Dempe in [15,16]. Authors in [17][18][19] extended a partial contraflow approach in multicommodity flow problems and provided the solution.…”

Dynamic Contraflow with Orientation Dependent Transit Times Allowing Intermediate Storage

“…The maximum dynamic contraflow (MDCF) problem with intermediate storage and its polynomial solution can be found in Pyakurel and Dempe [23]. By reverting only necessary arcs, Dhamala et al [5] presented algorithms to solve the quickest multi-commodity contraflow problem, where…”