In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (DCUT) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the DCUT formulation on metric instances. A key contribution of our work is extending of the Gap problem, previously explored in the context of the Traveling Salesman problems, to the metric Steiner Tree problem. To tackle the Gap problem for Steiner Tree instances, we first establish several structural properties of the CM formulation. We then classify the isomorphism classes of the vertices within the CM polytope, revealing a correspondence between the vertices of the DCUT and CM polytopes. Computationally, we exploit these structural properties to design two complementary heuristics for finding nontrivial small metric Steiner instances with a large integrality gap. We present several vertices for graphs with a number of nodes $\leq 10$, which realize the best-known lower bounds on the integrality gap for the CM and the DCUT formulations. We conclude the paper by presenting three new conjectures on the integrality gap of the DCUT and CM formulations for small graphs.

翻译：本文研究图上的度量斯坦纳树问题，重点计算双向割（DCUT）公式整数规划间隙的下界，并引入一种新颖的公式——完全度量（CM）模型，该模型专门针对度量实例上DCUT公式的弱点而设计。我们工作的一个关键贡献是将先前在旅行商问题背景下探讨的间隙问题扩展到度量斯坦纳树问题。为解决斯坦纳树实例的间隙问题，我们首先建立了CM公式的若干结构性质，进而对CM多面体中顶点的同构类进行分类，揭示了DCUT多面体与CM多面体顶点之间的对应关系。在计算层面，我们利用这些结构性质设计了两种互补的启发式算法，用于寻找具有较大整数规划间隙的非平凡小型度量斯坦纳树实例。我们给出了节点数 $\leq 10$ 的图的若干顶点，这些顶点实现了CM公式和DCUT公式整数规划间隙的当前最优下界。最后，我们针对小型图提出了关于DCUT和CM公式整数规划间隙的三个新猜想。