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.

翻译：Steiner树问题要求在一个无向图中以最低成本连接给定顶点子集。该问题最著名的线性规划松弛之一是双向割松弛（BCR）。确定该松弛的整数性间隙是一个长期存在的开放问题。数十年来，已知的最佳上界为2（可通过标准技术达到）。直到最近，Byrka、Grandoni和Traub [FOCS 2024] 才证明BCR的整数性间隙严格小于2。本文证明BCR的整数性间隙至多为1.898，显著改进了先前1.9988的界。对于终端最小生成树即为最优Steiner树的重要特例，我们通过对Byrka等人提出的对偶增长过程进行紧致分析，证明其整数性间隙至多为12/7。为获得1.898的通用界，我们将该对偶增长过程推广至更广泛的增长壁垒算法类。此外，我们证明此类增长壁垒算法无法产生能证明整数性间隙低于12/7的对偶解。最后，我们指出了该问题与超图松弛之间一个有趣的关联。