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.

翻译：大规模物流网络设计中的一个关键挑战是优化包裹分拣以实现高效路由。本文基于Van Dyk、Klause、Koenemann和Megow（IPCO 2024）近期提出的图论形式化框架研究该问题。该问题要求：给定输入有向图D（履约网络）及表示为源-汇对的一组物流需求，寻找D传递闭包中具有最小出度的子图H，使得该子图包含每个物流需求的源-汇路径。基于实际应用背景，我们研究了该问题的两个变体，区别在于物流需求的路径是预先给定还是可任意选择。我们对两个问题的复杂度进行了系统的参数化分析。研究结果聚焦于三个基本参数化方向：（1）当以H的目标出度为参数时，即使在高度受限的情况下，问题仍显示为paraNP困难；（2）以物流需求数量为参数时，我们运用拉姆齐型论证与着色编码技术为两个问题构建了参数化算法；（3）以D的结构为参数时，我们证明了两个问题关于树宽、最大度及最大路由长度的固定参数可解性。结合下界分析表明，若省略三个参数中的任意一个，问题将退化为paraNP困难。