A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least $36/31$, and it has long been conjectured that its true value is very close to this lower bound. However, the best upper bound for general graphs is still $2$. With the aim of circumventing the asymmetric nature of BCR, Chakrabarty, Devanur and Vazirani [Math. Program., 130 (2011), pp. 1--32] introduced the simplex-embedding LP, which is equivalent to it. Using this, they gave a $\sqrt{2}$-approximation algorithm for quasi-bipartite graphs and showed that the integrality gap of the relaxation is at most $4/3$ for this class of graphs. In this paper, we extend the approach provided by these authors and show that the integrality gap of BCR is at most $7/6$ on quasi-bipartite graphs via a fast combinatorial algorithm. In doing so, we introduce a general technique, in particular a potentially widely applicable extension of the primal-dual schema. Roughly speaking, we apply the schema twice with variable rates of growth for the duals in the second phase, where the rates depend on the degrees of the duals computed in the first phase. This technique breaks the disadvantage of increasing dual variables in a monotone manner and creates a larger total dual value, thus presumably attaining the true integrality gap.

翻译：改善施泰纳树近似算法的一个有希望的方法是使用双向削减放松(BCR),这一放松的整体性差距至少为36/31美元,而且长期以来人们一直推测,其真实值非常接近这一较低约束值。然而,一般图表的最大上限仍然是2美元。为了绕过BCR、Chakrabarty、Devanur和Vazirani的不对称性质[Math. program.,130(2011),pp.1-32],采用简单x混合的LP,这是相当的。利用这个方法,它们给出了半双部分图表的半部分整体性差距至少为36/31美元,而且长期以来一直认为其真实值非常接近。在本文件中,我们扩展了这些作者提供的方法,并表明,通过快速调控算算算算算算法,BCRCR的整体性差距最多为7/6美元。在这样做时,我们首先采用了一种总体性技术,因此将双部分的双向值推算法推算出一个潜在的双向性阶段。