Bridging the approximation gap or proving a hardness separation between polynomial and quasi-polynomial time for DST and GST

Determine whether the current approximation-ratio gap—O(log^3 k) achievable in polynomial time versus O(log^2 k / log log k) achievable in quasi-polynomial time—can be closed for Directed Steiner Tree or Group Steiner Tree, or else establish a lower bound that provably separates the approximation capabilities of polynomial-time and quasi-polynomial-time algorithms for these problems.

Background

The authors obtain a randomized polynomial-time O(log^3 k)-approximation for DST, which nearly matches the quasi-polynomial-time O(log^2 k / log log k) results. A similar gap is known for Group Steiner Tree (GST). This motivates the question of whether the extra logarithmic factor can be eliminated in polynomial time, or whether a complexity-theoretic barrier proves a strict separation between what polynomial-time and quasi-polynomial-time algorithms can achieve.

The authors explicitly flag this as an open question in their conclusion, framing future directions both in algorithm design (closing the gap) and in hardness (proving separation).