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Hierarchy of Hub Covering Problems

Niklas Jost

TL;DR

Hub Covering Problems (HCP) are threshold-constrained hub-location problems where tours between origin-destination pairs use at most two hubs. The paper defines three variants (1–3) in both single allocation (SA) and multi allocation (MA) forms, plus capacitated versions, and presents a formal metric-graph model with hub-opening costs and a threshold $\phi$ on tours or connections. The main contributions establish a hierarchy $MA3 \le_p MA2 \le_p MA1$ and $SA3 \le_p SA2 \le_p SA1$, show $SA/MA3$ is equivalent to Weighted Set Cover (hence $\Theta(\log|B|)$ lower bounds), and provide a $|B|^2$-approximation for $MA1$ (with a $|H|/k$ bound in the unweighted case). It also proves nonexistence of approximation guarantees for $SA1$, $SA2$, and capacitated HCPs, highlighting a sizable gap between lower and upper bounds and motivating Set Cover heuristics for practical solving. The work lays a formal foundation for comparing HCP variants and suggests heuristic directions based on Set Cover to tackle large-scale instances.

Abstract

Hub Covering Problems arise in various practical domains, such as urban planning, cargo delivery systems, airline networks, telecommunication network design, and e-mobility. The task is to select a set of hubs that enable tours between designated origin-destination pairs while ensuring that any tour includes no more than two hubs and that either the overall tour length or the longest individual edge is kept within prescribed limits. In literature, three primary variants of this problem are distinguished by their specific constraints. Each version exists in a single and multi allocation version, resulting in multiple distinct problem statements. Furthermore, the capacitated versions of these problems introduce additional restrictions on the maximum number of hubs that can be opened. It is currently unclear whether some variants are more complex than others, and no approximation bound is known. In this paper, we establish a hierarchy among these problems, demonstrating that certain variants are indeed special cases of others. For each problem, we either determine the absence of any approximation bound or provide both upper and lower bounds on the approximation guarantee.

Hierarchy of Hub Covering Problems

TL;DR

Hub Covering Problems (HCP) are threshold-constrained hub-location problems where tours between origin-destination pairs use at most two hubs. The paper defines three variants (1–3) in both single allocation (SA) and multi allocation (MA) forms, plus capacitated versions, and presents a formal metric-graph model with hub-opening costs and a threshold on tours or connections. The main contributions establish a hierarchy and , show is equivalent to Weighted Set Cover (hence lower bounds), and provide a -approximation for (with a bound in the unweighted case). It also proves nonexistence of approximation guarantees for , , and capacitated HCPs, highlighting a sizable gap between lower and upper bounds and motivating Set Cover heuristics for practical solving. The work lays a formal foundation for comparing HCP variants and suggests heuristic directions based on Set Cover to tackle large-scale instances.

Abstract

Hub Covering Problems arise in various practical domains, such as urban planning, cargo delivery systems, airline networks, telecommunication network design, and e-mobility. The task is to select a set of hubs that enable tours between designated origin-destination pairs while ensuring that any tour includes no more than two hubs and that either the overall tour length or the longest individual edge is kept within prescribed limits. In literature, three primary variants of this problem are distinguished by their specific constraints. Each version exists in a single and multi allocation version, resulting in multiple distinct problem statements. Furthermore, the capacitated versions of these problems introduce additional restrictions on the maximum number of hubs that can be opened. It is currently unclear whether some variants are more complex than others, and no approximation bound is known. In this paper, we establish a hierarchy among these problems, demonstrating that certain variants are indeed special cases of others. For each problem, we either determine the absence of any approximation bound or provide both upper and lower bounds on the approximation guarantee.

Paper Structure

This paper contains 4 sections, 6 theorems, 14 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Model Formulation
  3. Proofs
  4. Conclusion

Key Result

Lemma 1

MA 1 can be $|B|^2$ approximated. In the unweighted case, it can additionally be $\frac{|H|}{k}$ approximated for any constant $k$.

Figures (16)

  • Figure 1: Approximation guarantees for each variant under the assumption $P\neq NP$ shown in this paper.
  • Figure 2: The references show, which lemma describes the corresponding bound or reduction.
  • Figure 3: Example SA/MA 2 Problem
  • Figure 4: Transformed to SA/MA 1, black edges have weight 1 and blue edges have weight 2
  • Figure 5: Example 1: The dotted tour is valid and uses one hub.
  • ...and 11 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6