Hierarchy of Hub Covering Problems

2025-03-04

AI Summary

This paper studies the Hub Covering Problem (HCP), which seeks a minimum-size hub set such that every specified origin–destination (OD) pair is connected via a path traversing at most two hubs, with total or single-edge length bounded by a given threshold—arising in urban planning, logistics, air transport, and telecommunications. We systematically establish a rigorous hierarchy of six fundamental HCP variants—including single- and multiple-allocation models, with and without capacity constraints—clarifying their logical implications and computational complexity distinctions. Via novel reductions and lower-bound constructions, we fully characterize approximability: four variants are proven inapproximable within any constant factor, while two admit tight approximation bounds—i.e., optimal constant-factor approximations. Our work unifies modeling across variants, reveals intrinsic structural differences, and resolves long-standing open questions regarding approximation thresholds for HCP.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Classify complexity of Hub Covering Problems variants.
• Establish hierarchy among Hub Covering Problems variants.
• Determine approximation bounds for Hub Covering Problems.

Innovation

• Established hierarchy among hub covering problems
• Determined approximation bounds for variants
• Analyzed single and multi allocation versions