Polynomial-time polylogarithmic (or sub-polylogarithmic) approximation for Directed Steiner Tree

Determine whether there exists a polynomial-time algorithm that achieves a polylogarithmic approximation ratio for the Directed Steiner Tree problem on arbitrary directed graphs; furthermore, determine whether any polynomial-time algorithm can achieve a sub-polylogarithmic approximation ratio for Directed Steiner Tree.

Background

The Directed Steiner Tree (DST) problem seeks a minimum-cost subgraph of a directed graph that connects a designated root to all terminals via directed paths. Prior to this work, the best-known polynomial-time approximations achieved only k^ε for any fixed ε>0, while quasi-polynomial-time algorithms attained polylogarithmic ratios such as O(log^3 k) and O(log^2 k / log log k). This longstanding gap led to a central question about whether polylogarithmic approximations could be achieved in polynomial time for DST.

The paper states this open problem explicitly in the introduction and then presents an O(log^3 k) polynomial-time approximation algorithm, thereby resolving the first part of the question. The statement here records the historically central problem as it was explicitly posed by the authors.