The Bidirected Cut Relaxation for Steiner Tree: Better Integrality Gap Bounds and the Limits of Moat Growing

2026-02-23

AI Summary

This study addresses the long-standing open problem of the integrality gap of the Bidirected Cut Relaxation (BCR) for the Steiner tree problem. By generalizing the dual-fitting primal-dual growth process into a broader “moat-growing algorithm” framework and combining linear programming duality with a tightness analysis, the authors significantly improve the upper bound on the integrality gap in the general case to 1.898. Furthermore, they prove that any moat-growing algorithm cannot achieve a performance ratio better than 12/7 in the special case where the minimum spanning tree over terminals is itself an optimal Steiner tree. This limitation is shown to be intrinsically linked to the structure of hypergraph relaxations, revealing a deep connection between algorithmic barriers and extended formulations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Steiner Tree problem asks for the cheapest way of connecting a given subset of the vertices in an undirected graph. One of the most prominent linear programming relaxations for Steiner Tree is the Bidirected Cut Relaxation (BCR). Determining the integrality gap of this relaxation is a long-standing open question. For several decades, the best known upper bound was 2, which is achievable by standard techniques. Only very recently, Byrka, Grandoni, and Traub [FOCS 2024] showed that the integrality gap of BCR is strictly below 2. We prove that the integrality gap of BCR is at most 1.898, improving significantly on the previous bound of 1.9988. For the important special case where a terminal minimum spanning tree is an optimal Steiner tree, we show that the integrality gap is at most 12/7, by providing a tight analysis of the dual-growth procedure by Byrka et al. To obtain the general bound of 1.898 on the integrality gap, we generalize their dual growth procedure to a broad class of moat-growing algorithms. Moreover, we prove that no such moat-growing algorithm yields dual solutions certifying an integrality gap below 12/7. Finally, we observe an interesting connection to the Hypergraphic Relaxation.

Problem

• Steiner Tree
• Bidirected Cut Relaxation
• Integrality Gap
• Moat Growing
• Linear Programming Relaxation

Innovation

• Bidirected Cut Relaxation
• Integrality Gap
• Moat-Growing Algorithm
• Steiner Tree
• Dual-Fitting