Train Marshalling Is Fixed Parameter Tractable

Abstract

Abstract. The train marshalling problem is about reordering the cars of a train using as few auxiliary rails as possible. The problem is known to be NP-complete. We show that it is fixed parameter tractable (FPT) with the number of auxiliary rails as parameter.

A Novel Dynamic Programming Approach to the Train Marshalling Problem

“…It has also be shown that the competitive factor of 2 is indeed best possible among all deterministic online algorithms. In [26], Brueggeman et al established that the TMP is fixed parameter tractable with respect to the number of classification tracks k. To be more precise, if an inbound train T with n railcars having t different arXiv:1903.08364v1 [cs.DS] 20 Mar 2019 c 2019 IEEE. Personal use of this material is permitted.…”

“…Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. destinations, and a positive integer k are given, then for deciding whether or not the railcars of T can be rearranged in an appropriate order by using at most k classification tracks, the algorithm proposed in [26] requires O(2 O(k) poly(n)) time and O(n 2 k2 8k ) space. The algorithm is based on the dynamic programming (DP) paradigm.…”

“…DP is indeed an approach of great importance in the design of both polynomial-time and exponential-time algorithms [14]- [17]. In contrast to the exact methods described in [5], [26], our method is developed to directly solve the optimization version of the TMP (i.e., it does not need to make successive calls to a procedure that solves the decision version of the problem). 1 The proposed algorithm is capable of finding not only the minimum number of required classification tracks, but also the classification track that each railcar is assigned to, in an optimal solution, as well.…”

“…It has also be shown that the competitive factor of 2 is indeed best possible among all deterministic online algorithms. In [26], Brueggeman et al established that the TMP is fixed parameter tractable with respect to the number of classification tracks k. To be more precise, if an inbound train T with n railcars having t different destinations, and a positive integer k are given, then for deciding whether or not the railcars of T can be rearranged in an appropriate order by using at most k classification tracks, the algorithm proposed in [26] requires O(2 O(k) poly(n)) time and O(n 2 k2 8k ) space. The algorithm is based on the dynamic programming (DP) paradigm.…”

A Graph-Theoretic Approach to the Train Marshalling Problem

“…In 2000, Dahlhaus et al [6] proved that TMP is N P-complete and introduced new bounds. Brueggeman et al show in [7] that the problem is fixed parameter tractable. In another work by Dahlhaus, Manne, Miller and Ryan [8] they described similar problems.…”