The bidirected cut relaxation is the characteristic representative of the bidirected relaxations ($\mathrm{\mathcal{BCR}}$) which are a well-known class of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs (STP). Although no general approximation algorithm based on $\mathrm{\mathcal{BCR}}$ with an approximation ratio better than $2$ for STP is known, it is mostly preferred in integer programming as an implementation of STP, since there exists a formulation of compact size, which turns out to be very effective in practice. It is known that the integrality gap of $\mathrm{\mathcal{BCR}}$ is at most $2$, and a long standing open question is whether the integrality gap is less than $2$ or not. The best lower bound so far is $\frac{36}{31} \approx 1.161$ proven by Byrka et al. [BGRS13]. Based on the work of Chakrabarty et al. [CDV11] about embedding STP instances into simplices by considering appropriate dual formulations, we improve on this result by constructing a new class of instances and showing that their integrality gaps tend at least to $\frac{6}{5} = 1.2$. More precisely, we consider the class of equivalent LP-relaxations $\mathrm{\mathcal{BCR}}^{+}$, that can be obtained by strengthening $\mathrm{\mathcal{BCR}}$ by already known straightforward Steiner vertex degree constraints, and show that the worst case ratio regarding the optimum value between $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{BCR}}^{+}$ is at least $\frac{6}{5}$. Since $\mathrm{\mathcal{BCR}}^{+}$ is a lower bound for the hypergraphic relaxations ($\mathrm{\mathcal{HYP}}$), another well-known class of equivalent LP-relaxations on which the current best $(\ln(4) + \varepsilon)$-approximation algorithm for STP by Byrka et al. [BGRS13] is based, this worst case ratio also holds for $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{HYP}}$.

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基于单纯形的 Steiner 树实例为双向切割松弛产生很大的完整性差距

双向切割松弛是双向松弛 ($\mathrm{\mathcal{BCR}}$) 的特征代表，它是 NP-hard Steiner Tree Problem in Graphs (STP) 的著名等效 LP 松弛类. 尽管没有已知的基于 $\mathrm{\mathcal{BCR}}$ 的近似比值优于 $2$ 的 STP 的通用逼近算法，但它在整数规划中作为 STP 的实现最受青睐，因为存在以下公式紧凑的尺寸，这在实践中非常有效。众所周知，$\mathrm{\mathcal{BCR}}$ 的完整性差距最多为 $2$，而一个长期悬而未决的问题是完整性差距是否小于 $2$。迄今为止最好的下限是 $\frac{36}{31} \approx 1.161$ 由 Byrka 等人证明。[BGRS13]。基于 Chakrabarty 等人的工作。[CDV11] 关于通过考虑适当的对偶公式将 STP 实例嵌入到单纯形中，我们通过构建一类新的实例来改进这个结果，并表明它们的完整性差距至少趋于 $\frac{6}{5} = 1.2$。更准确地说，我们考虑等价 LP 松弛 $\mathrm{\mathcal{BCR}}^{+}$ 的类，可以通过已知的直接 Steiner 加强 $\mathrm{\mathcal{BCR}}$ 获得顶点度约束，并表明关于 $\mathrm{\mathcal{BCR}}$ 和 $\mathrm{\mathcal{BCR}}^{+}$ 之间的最佳值的最坏情况比率至少为 $\frac{ 6}{5}$。由于 $\mathrm{\mathcal{BCR}}^{+}$ 是超图形松弛的下限 ($\mathrm{\mathcal{HYP}}$)，另一类众所周知的等效 LP 松弛，Byrka 等人在该类上提出了 STP 的当前最佳 $(\ln(4) + \varepsilon)$-近似算法。[BGRS13] 是基于，这个最坏情况比率也适用于 $\mathrm{\mathcal{BCR}}$ 和 $\mathrm{\mathcal{HYP}}$。