The Directed Steiner Tree (DST) problem is defined on a directed graph $G=(V,E)$, where we are given a designated root vertex $r$ and a set of $k$ terminals $K \subseteq V \setminus {r}$. The goal is to find a minimum-cost subgraph that provides directed $r \rightarrow t$ paths for all terminals $t \in K$. The approximability of DST has long been a central open problem in network design. While there exist polylogarithmic-approximation algorithms with quasi-polynomial running times (Charikar et al. 1998; Grandoni, Laekhanukit, and Li 2019; Ghuge and Nagarajan 2020), the best known polynomial-time approximation until now has remained at $k^\epsilon$, for any constant $\epsilon > 0$. Whether a polynomial-time algorithm achieving a polylogarithmic approximation exists has remained unresolved. In this paper, we present a flow-based LP-relaxation for DST that admits a polylogarithmic integrality gap under the relative integral condition -- there exists a fractional solution in which each edge $e$ either carries a zero flow ($f^t_e=0$) or uses its full capacity ($f^t_e=x_e$), where $f^t_e$ denotes the flow variable and $x_e$ denotes the indicator variable treated as capacities. This stands in contrast to known lower bounds, as the standard flow-based relaxation is known to exhibit a polynomial integrality gap even under relatively integral solutions. In fact, this relatively integral property is shared by all the known integrality gap instances of DST [Halperin~et~al., SODA'07; Zosin-Khuller, SODA'02; Li-Laekhanukit, SODA'22]. We further provide a randomized polynomial-time algorithm that gives an $O(\log^3 k)$-approximation, assuming access to a relatively integral fractional solution.

翻译：有向斯坦纳树（DST）问题定义在有向图 $G=(V,E)$ 上，其中给定一个指定根顶点 $r$ 和一个包含 $k$ 个终端的集合 $K \subseteq V \setminus {r}$。目标是找到一个最小成本的子图，该子图为所有终端 $t \in K$ 提供从 $r$ 到 $t$ 的有向路径。DST的近似性长期以来一直是网络设计中的一个核心开放问题。虽然存在具有拟多项式运行时间的多对数近似算法（Charikar 等人，1998；Grandoni、Laekhanukit 和 Li，2019；Ghuge 和 Nagarajan，2020），但迄今为止已知的最佳多项式时间近似比对于任意常数 $\epsilon > 0$ 仍为 $k^\epsilon$。是否存在一个能实现多对数近似的多项式时间算法仍未解决。在本文中，我们为DST提出了一种基于流的线性规划松弛，该松弛在相对整数条件下允许一个多对数的整数性间隙——即存在一个分数解，其中每条边 $e$ 要么承载零流（$f^t_e=0$），要么使用其全部容量（$f^t_e=x_e$），这里 $f^t_e$ 表示流变量，$x_e$ 表示被视为容量的指示变量。这与已知的下界形成对比，因为已知标准的基于流的松弛即使在相对整数解下也会表现出多项式的整数性间隙。事实上，所有已知的DST整数性间隙实例都具有这种相对整数性质 [Halperin 等人，SODA'07；Zosin-Khuller，SODA'02；Li-Laekhanukit，SODA'22]。我们进一步提供了一个随机多项式时间算法，该算法在假设能够获得一个相对整数分数解的情况下，能够给出一个 $O(\log^3 k)$ 的近似比。