In the Directed Steiner Tree (DST) problem, we are given a directed graph $G=(V,E)$ on $n$ vertices with edge-costs $c \in \mathbb{R}_{\geq 0}^E$, a root vertex $r$, and a set $K$ of $k$ terminals. The goal is to find a minimum-cost subgraph of $G$ that contains a path from $r$ to every terminal $t \in K$. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of $O(k^\epsilon)$ for any constant $\epsilon > 0$, and the best quasi-polynomial-time approximation ratio of $O\left(\frac{\log^2 k}{\log \log k}\right)$. Towards understanding this gap, we study the integrality gap of the standard flow LP relaxation for the problem. We show that the LP has an integrality gap polynomial in $n$. Previously, the integrality gap LP is only known to be $\Omega\left(\frac{\log^2n}{\log\log n}\right)$ [Halperin et al., SODA'03 \& SIAM J. Comput.] and $\Omega(\sqrt{k})$ [Zosin-Khuller, SODA'02] in some instance with $\sqrt{k}=O\left(\frac{\log n}{\log \log n}\right)$. Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in $n$, the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.
翻译:在直接的 Steiner 树树( DST) 问题中, 我们得到一个直线的图表 $G = (V, E) 。 数十年来, DST一直是一个臭名昭著的问题, 因为任何恒定的 $\ epbb{Rägeq 0 $E$, 根顶价为 $K美元, 以及固定的 $O 美元 美元 的 美元 。 目标是找到一个最低成本的 $G 的子图, 包含一条从美元到每个终端$的路径 。 在第一个缺口上, 我们研究了标准流动的内基( 美元) 的内基值 。 我们显示, LP = $( k) 的内基值是 美元内基值 。