Johnson Phosavanh | Searcharxiv

arXiv2026

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

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.

cs.DS↗

arXiv2024

Centrality of shortest paths: Algorithms and complexity results

The degree centrality of a node, defined as the number of nodes adjacent to it, is often used as a measure of importance of a node to the structure of a network. This metric can be extended to paths in a network, where the degree centrality of a path is defined as the number of nodes adjacent to it. In this paper, we reconsider the problem of finding the most degree-central shortest path in an unweighted network. We propose a polynomial algorithm with the worst-case running time of $O(|E||V|^2\Delta(G))$, where $|V|$ is the number of vertices in the network, $|E|$ is the number of edges in the network, and $\Delta(G)$ is the maximum degree of the graph. We conduct a numerical study of our algorithm on synthetic and real-world networks and compare our results to the existing literature. In addition, we show that the same problem is NP-hard when a weighted graph is considered. Furthermore, we consider other centrality measures, such as the betweenness and closeness centrality, showing that the problem of finding the most betweenness-central shortest path is solvable in polynomial time and finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not.

cs.DS↗