The Steiner Shortest Path Tree Problem

2025-09-08

AI Summary

This paper formally defines and studies the Steiner Shortest Path Tree (SSPT) problem: given a directed graph, a source vertex, and a set of terminal vertices, compute a shortest-path tree rooted at the source that spans all terminals while minimizing the number of non-terminal (Steiner) vertices. SSPT is NP-hard and models network connectivity scenarios where intermediate-node overhead must be minimized. The authors introduce a “shortest-path subgraph” construction technique to reduce SSPT to the uniformly weighted Directed Steiner Tree (DST) problem, preserving approximation ratios. Leveraging this reduction, they design a quasi-polynomial-time $O(log^2 k log n)$-approximation algorithm, where $k$ is the number of terminals and $n$ the number of vertices. Furthermore, for restricted graph classes—including those with bounded treewidth—they obtain polynomial-time polylogarithmic approximation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce and study a novel problem of computing a shortest path tree with a minimum number of non-terminals. It can be viewed as an (unweighted) Steiner Shortest Path Tree (SSPT) that spans a given set of terminal vertices by shortest paths from a given source while minimizing the number of nonterminal vertices included in the tree. This problem is motivated by applications where shortest-path connections from a source are essential, and where reducing the number of intermediate vertices helps limit cost, complexity, or overhead. We show that the SSPT problem is NP-hard. To approximate it, we introduce and study the shortest path subgraph of a graph. Using it, we show an approximation-preserving reduction of SSPT to the uniform vertex-weighted variant of the Directed Steiner Tree (DST) problem, termed UVDST. Consequently, the algorithm of [Grandoni et al., 2023] approximating DST implies a quasi-polynomial polylog-approximation algorithm for SSPT. We present a polynomial polylog-approximation algorithm for UVDST, and thus for SSPT, for a restricted class of graphs.

Problem

• Minimizing non-terminal vertices in shortest path trees
• Solving NP-hard Steiner Shortest Path Tree problem
• Approximating solution for restricted graph classes

Innovation

• Approximates SSPT via reduction to UVDST problem
• Uses quasi-polynomial polylog-approximation algorithm approach
• Develops polynomial algorithm for restricted graph classes