A simple Path-based LP Relaxation for Directed Steiner Tree

2026-03-03

AI Summary

This work proposes a path-based linear programming relaxation for the Directed Steiner Tree (DST) problem on layered graphs, circumventing the conventional lift-and-project hierarchy. The approach achieves the current best-known integrality gap bound of \(O(l \log k)\) for graphs with \(k\) terminals and \(l\) layers in a more transparent and streamlined manner. Furthermore, it demonstrates that only \(O(l)\) rounds of the Sherali-Adams hierarchy are sufficient to significantly narrow the integrality gap, while also simplifying the proof of convergence for this hierarchy. By integrating path-based relaxations, combinatorial optimization insights, and integer programming analysis, this study establishes a more efficient and theoretically clear relaxation framework for the DST problem.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the Directed Steiner Tree (DST) problem in layered graphs through a simple path-based linear programming relaxation. This relaxation achieves an integrality gap of O(l log k), where k is the number of terminals and l is the number of layers, which matches the best known bounds for DST previously obtained via lift-and-project hierarchies. Our formulation bypasses hierarchy machinery, offering a more transparent route to the state-of-the-art bound, and it can be exploited to provide an alternative simpler proof that O(l) rounds of the Sherali-Adams hierarchy suffice for reducing the integrality gap on layered instances of DST.

Problem

• Directed Steiner Tree
• layered graphs
• integrality gap
• linear programming relaxation

Innovation

• Directed Steiner Tree
• LP Relaxation
• Integrality Gap
• Sherali-Adams Hierarchy
• Layered Graphs