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

TL;DR

This paper improves the upper bound on the integrality gap of the Bidirected Cut Relaxation for Steiner Tree from 1.9988 to 1.898, and explores the limits of moat-growing algorithms in this context.

Contribution

It provides a tighter bound on the integrality gap of BCR and analyzes the limitations of moat-growing algorithms for Steiner Tree relaxations.

Findings

Integrality gap of BCR is at most 1.898.

Special case gap bound is 12/7.

Moat-growing algorithms cannot certify gaps below 12/7.

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.