In this work, we compute the lower bound of the integrality gap of the Metric Steiner Tree Problem (MSTP) on a graph for some small values of number of nodes and terminals. After debating about some limitations of the most used formulation for the Steiner Tree Problem, namely the Bidirected Cut Formulation, we introduce a novel formulation, that we named Complete Metric formulation, tailored for the metric case. We prove some interesting properties of this formulation and characterize some types of vertices. Finally, we define a linear program (LP) by adapting a method already used in the context of the Travelling Salesman Problem. This LP takes as input a vertex of the polytope of the CM relaxation and provides an MSTP instance such that (a) the optimal solution is precisely that vertex and (b) among all of the instances having that vertex as its optimal solution, the selected instance is the one having the highest integrality gap. We propose two heuristics for generating vertices to provide inputs for our procedure. In conclusion, we raise several conjectures and open questions.

翻译：在本研究中，我们针对节点数与终端数较小的若干情况，计算了图上度量斯坦纳树问题（MSTP）整数规划间隙的下界。在探讨了斯坦纳树问题最常用公式——即有向割公式——的若干局限性后，我们提出了一种专为度量情形设计的新颖公式，并将其命名为完全度量公式。我们证明了该公式的一些重要性质，并对特定类型的顶点进行了刻画。最后，我们通过改编旅行商问题中已使用的方法定义了一个线性规划（LP）。该线性规划以完全度量松弛多面体的一个顶点作为输入，并输出一个满足以下条件的MSTP实例：（a）该实例的最优解恰好是该顶点；（b）在所有以该顶点作为最优解的实例中，所选实例具有最大的整数规划间隙。我们提出了两种启发式算法来生成顶点，以作为我们方法的输入数据。最后，我们提出了若干猜想与待解问题。