Designing Efficient and Reachable Routes: The $k$-Step-Central Shortest Path Problem

2026-06-12

AI Summary

This work proposes the $k$-step central shortest path problem, for which it designs a polynomial-time algorithm on unweighted graphs augmented with a novel pruning strategy, while proving the weighted variant to be NP-hard. The study introduces a new measure of reachability—quantified by the number of nodes adjacent to a path—and integrates it into shortest path optimization to jointly enhance path efficiency and coverage. The proposed framework also solves the closeness-central shortest path problem, which is shown to be NP-hard. Extensive experiments on synthetic and real-world networks with up to 2,000 nodes demonstrate the algorithm’s efficiency and scalability, revealing that the approach significantly improves coverage compared to strategies that merely extend the backbone path length.

Abstract

Designing rapid transportation routes requires balancing efficiency and reachability. Shortest-path models ensure direct, cost-efficient routes but ignore coverage, while centrality-based approaches maximize accessibility but do not enforce operational constraints. We study the problem of selecting a shortest path that maximizes reachability, measured as the number of nodes within a fixed distance of the path. To do this, we introduce the $k$-Step-Central Shortest Path problem and analyse its structural properties. We show that optimal solutions on unweighted graphs can be found in polynomial time and propose an algorithm with a novel pruning rule. We also prove that the problem becomes NP-hard when edge weights are introduced. Additionally, we show that our algorithm can be used to solve the NP-hard problem of finding the closeness-central shortest path in a graph. We demonstrate the efficiency and scalability of our algorithm on synthetic and real-world networks with up to 2,000 nodes. Our results show that improving reachability can substitute for route expansion: increasing the reach of transit lines drastically increases their coverage with shorter routes. This suggests that investments in active transport infrastructure that improve reachability can be more effective than extending primary routes, providing a data-driven basis for allocating resources in network design.

Problem

• shortest path
• reachability
• centrality
• network design
• k-step-central

Innovation

• k-Step-Central Shortest Path
• reachability optimization
• polynomial-time algorithm
• NP-hardness
• pruning rule