Parameterized Complexity of Efficient Sortation

TL;DR

The paper studies parcel sortation by modeling two graph routing problems, MD-SPP and MD-RSPP, on a digraph $D$ with commodities, seeking a low-outdegree subgraph of ${\cal T}(D)$ that supports all required routes. It provides a comprehensive parameterized complexity map across three axes: the target outdegree $T$, the number of commodities $|K|$, and structural properties of $D$ (treewidth, degree) together with a routing length bound $p$. Key results include paraNP-hardness for the target-based parameter, FPT algorithms when parameterizing by the number of commodities (via kernelization, Ramsey-type arguments, and color-coding), and FPT algorithms for structural parameters when the maximum path length is bounded (via dynamic programming on tree decompositions). The work also introduces SMD-SPP as a conceptual tool and demonstrates that a combined parameterization $(tw,\Delta,p)$ yields tractability, while dropping any parameter leads to para-hardness, thereby charting a nuanced landscape for practical routing in large-scale logistics. Overall, the paper provides a detailed roadmap of when these two routing problems are tractable and how to exploit their structure in real-world networks.

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.

Figures (1)

• Figure 1: Example network for MD-SPP and MD-RSPP. The left-most image shows an input graph $D$. Five commodities for MD-SPP, $(v_1,v_2, (v_1,v_3,v_6,v_4,v_2)), (v_2,v_5,(v_2,v_3,v_5)), (v_3,v_2,(v_3,v_4,v_2)), (v_3,v_4,(v_3,v_4)), (v_3,v_6,(v_3,v_6))$, are represented in the second image. The third image gives a minimum target $(T=2)$ solution graph $H$ to the resulting MD-SPP instance. The last image illustrates that for MD-RSPP, by not fixing the paths of commodities, solutions $H$ with smaller target might exist (here $T=1$). Note that the multi-edges are only for illustration. A solution graph $H$ is in fact simple.

Theorems & Definitions (12)

• Theorem 2
• Theorem 3
• Lemma 4
• Theorem 5
• Lemma 7
• Definition 8: $q$-enclosure
• Lemma 9
• Theorem 10
• Theorem 11
• Theorem 12