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, kernelization and treewidth reduction 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进行路由。我们基于Van Dyk、Klause、Koenemann和Megow（IPCO 2024）最近提出的图论形式化框架研究该问题。该问题要求：给定一个输入有向图D（履约网络）以及一组以源-汇元组表示的货物，在D的传递闭包中寻找一个最小出度的子图H，使得H包含每条货物的源-汇路径。鉴于其底层动机，我们研究了该问题的两种变体，其区别在于货物的路径是假设已给定还是可任意选择。我们对两个问题的复杂度进行了彻底的参数化分析。我们的研究聚焦于问题的三种基本参数化方式：（1）当尝试以H的目标出度作为参数时，我们证明即使在高度受限的情形下，这些问题也是paraNP困难的；（2）当以货物数量作为参数时，我们利用拉姆齐型论证、核化技术和树宽约简技术，为两个问题设计了参数化算法；（3）当以D的结构作为参数时，我们证明了两个问题关于树宽、最大度和最大路由长度均具有固定参数可解性。结合下界分析，我们表明若省略这三个参数中的任意一个，问题将导致paraNP困难性。