🤖 AI Summary
Stacker Crane Problem (SCP) seeks shortest tours on a graph to serve multiple pickup-delivery pairs. SCP is NP-hard on general graphs—including trees—while polynomial-time algorithms exist only for paths and cycles. This paper establishes, for the first time, that SCP is fixed-parameter tractable (FPT) with respect to both cyclomatic number and number of branch vertices. Leveraging this result, we design a near-linear-time optimal algorithm for arbitrary graphs of fixed topology. Our approach integrates structural graph decomposition, dynamic programming, and systematic tree/cycle expansion techniques. It unifies and generalizes prior work by delivering polynomial-time optimal solutions not only on trees and cycles but also on broader classes of graphs with bounded topological complexity. This breaks the long-standing limitation of existing algorithms, which were restricted to trivial topologies.
📝 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 tree structures. 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.