论文标题
基于单纯形的施泰纳树实例产生了较大的完整差距
Simplex based Steiner tree instances yield large integrality gaps for the bidirected cut relaxation
论文作者
论文摘要
双向切割放松是bidectioned放松的特征代表（$ \ mathrm {\ mathcal {bcr}} $），它是图形中NP-Hard-Hard Stoiner树问题（STP）的NP-Hard-Hard Stoiner Tree问题（STP）。尽管没有基于$ \ mathrm {\ Mathcal {bcr}} $具有近似值比$ 2 $的一般近似算法，但它在整数编程中最喜欢STP的实现，因为在实践中存在非常有效的紧凑型尺寸，因此它具有非常有效的尺寸。 众所周知，$ \ mathrm {\ mathcal {bcr}} $的整体差距最多是$ 2 $，一个长期的开放问题是，整数差距是否少于$ 2 $。到目前为止，最好的下限是$ \ frac {36} {31} \大约1.161 $由Byrka等人证明。 [BGRS13]。根据Chakrabarty等人的工作。 [CDV11]关于通过考虑适当的双重公式将STP实例嵌入简单中，我们通过构建新的一类实例来改进结果，并表明它们的完整性差距至少趋向于$ \ frac {6} {5} {5} = 1.2 $。 更准确地说，我们考虑了等效的LP-RELAXITATION $ \ MATHRM {\ MATHCAL {bcr}}}^{+} $，可以通过加强$ \ Mathrm {\ Mathrm {\ Mathcal {bcr {bcr}} $来获得已知的直接施加式condertex condection word contigion word contemion word contime for tabinection contection and Condection。 $ \ mathrm {\ Mathcal {bcr}} $和$ \ mathrm {\ Mathcal {bcr}}}}^{+} $至少是$ \ frac {6} {5} $。由于$ \ mathrm {\ Mathcal {bcr}}^{ +} $是超图放松的下限（$ \ m artrm {\ mathrm {\ mathcal {help}} $），这是另一种众所周知的等效lp-relaxations类别的最佳$（4） + \ ln（4） + \ v byepsmim- forpsmim forpsim forpsim forpim- forepsim forepsim- Byrka等。 [BGRS13]是基于$ \ Mathrm {\ Mathcal {bcr}} $和$ \ Mathrm {\ Mathcal {hyp {help}} $的最坏情况比率的基础。
The bidirected cut relaxation is the characteristic representative of the bidirected relaxations ($\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 $\mathrm{\mathcal{HYP}}$.
