An Optimal Algorithm for the Stacker Crane Problem on Fixed Topologies

TL;DR

This paper introduces an optimal, near-linear time algorithm for the Stacker Crane Problem on fixed graph topologies, leveraging fixed-parameter tractability to efficiently solve a complex NP-hard routing challenge.

Contribution

The paper presents a novel algorithm that is optimal for fixed topologies, expanding the class of efficiently solvable instances of the SCP.

Findings

Algorithm is optimal for each fixed topology.

Runs in near-linear time due to fixed-parameter tractability.

Applicable to graphs with fixed cycle rank and branch vertices.

Abstract

The Stacker Crane Problem (SCP) is a variant of the Traveling Salesman Problem. In SCP, pairs of pickup and delivery points are designated on a graph, and a crane must visit these points to move objects from each pickup location to its respective delivery point. The goal is to minimize the total distance traveled. SCP is known to be NP-hard, even on trees. The only positive results, in terms of polynomial-time solvability, apply to graphs that are topologically equivalent to a path or a cycle. We propose an algorithm that is optimal for each fixed topology, running in near-linear time. This is achieved by demonstrating that the problem is fixed-parameter tractable (FPT) when parameterized by both the cycle rank and the number of branch vertices.