Core Concepts
Optimizing parcel sortation in large-scale logistical networks by minimizing the number of sort points (outdegree) required in a subgraph of the transitive closure of the input network.
Abstract
The content discusses the problem of optimizing parcel sortation in large-scale logistical networks. It introduces two variants of the problem, the Min-Degree Sort Point Problem (MD-SPP) and the Min-Degree Routing and Sort Point Problem (MD-RSPP), which aim to find a subgraph of the transitive closure of the input network with minimum outdegree that satisfies the routing requirements of a set of commodities.
The key highlights and insights are:
- MD-SPP assumes the routing paths for the commodities are given, while MD-RSPP allows the paths to be chosen arbitrarily.
- The authors perform a thorough parameterized complexity analysis of both problems, considering three fundamental parameterizations: the target outdegree, the number of commodities, and the structural properties of the input graph.
- For the target outdegree parameterization, the problems are shown to be NP-hard even in highly restricted cases.
- When parameterizing by the number of commodities, the authors develop fixed-parameter algorithms for both problems, utilizing techniques such as Ramsey-type arguments, kernelization, and treewidth reduction.
- For the structural parameterization, the authors establish fixed-parameter tractability for both problems with respect to treewidth, maximum degree, and maximum routing length, and provide matching lower bounds.
Stats
The input graph D has n vertices and m edges.
The number of commodities is denoted by |K|.
The target outdegree is denoted by T.
Quotes
"The task of finding optimal solutions to logistical challenges has motivated the study of a wide range of computational graph problems including, e.g., the classical Vertex and Edge Disjoint Paths [25, 24, 18, 17] problems and Coordinated Motion Planning (also known as Multiagent Pathfinding) [22, 34, 20, 12]."
"When dealing with logistical challenges at a higher scale, collision avoidance (which is the main goal in the aforementioned two problems) is no longer relevant and one needs to consider different factors when optimizing or designing a logistical network."