Parameterized Complexity of Efficient Sortation

TL;DR

This paper analyzes the computational complexity of optimizing parcel sortation in large logistics networks, providing parameterized algorithms and hardness results for different problem variants and parameters.

Contribution

It offers a comprehensive parameterized complexity analysis of the graph-theoretic parcel sortation problem, including algorithms and hardness results based on various parameters.

Findings

Problems are paraNP-hard when parameterized by target outdegree.

Parameterized algorithms are developed for the number of commodities.

Fixed-parameter tractability is established for treewidth, degree, and routing length.

Abstract

A crucial challenge arising in the design of large-scale logistical networks is to optimize parcel sortation for routing. We study this problem under the recent graph-theoretic formalization of Van Dyk, Klause, Koenemann and Megow (IPCO 2024). The problem asks - given an input digraph D (the fulfillment network) together with a set of commodities represented as source-sink tuples - for a minimum-outdegree subgraph H of the transitive closure of D that contains a source-sink route for each of the commodities. Given the underlying motivation, we study two variants of the problem which differ in whether the routes for the commodities are assumed to be given, or can be chosen arbitrarily. We perform a thorough parameterized analysis of the complexity of both problems. Our results concentrate on three fundamental parameterizations of the problem: (1) When attempting to parameterize by the target outdegree of H, we show that the problems are paraNP-hard even in highly restricted cases; (2) When parameterizing by the number of commodities, we utilize Ramsey-type arguments and color-coding techniques to obtain parameterized algorithms for both problems; (3) When parameterizing by the structure of D, we establish fixed-parameter tractability for both problems w.r.t. treewidth, maximum degree and the maximum routing length. We combine this with lower bounds which show that omitting any of the three parameters results in paraNP-hardness.