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：本文针对节点与终端数量较小的情形，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公式，称之为完全度量公式。我们证明了该公式的若干重要性质，并对特定类型的顶点进行了表征。最后，通过借鉴旅行商问题中已有的方法，我们构建了一个线性规划模型。该模型以完全度量松弛多面体的顶点作为输入，输出一个满足以下条件的MSTP实例：（a）该顶点恰好是实例的最优解；（b）在所有以该顶点为最优解的实例中，所选实例具有最大的整数规划间隙。我们提出了两种生成顶点的启发式方法作为该流程的输入。最后，本文提出了若干猜想与待解问题。