On the Integrality Gap of Directed Steiner Tree LPs with Relatively Integral Solutions

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.