Improved Inapproximability Results for Steiner Tree via Long Code Based Reductions

The best algorithm for approximating Steiner tree has performance ratio $\ln(4)+\epsilon \approx 1.386$ [J. Byrka et al., \textit{Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC)}, 2010, pp. 583-592], whereas the inapproximability result stays at the factor $\frac{96}{95} \approx 1.0105$ [M. Chleb\'ik and J. Chleb\'ikov\'a, \textit{Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT)}, 2002, pp. 170-179]. In this article, we take a step forward to bridge this gap and show that there is no polynomial time algorithm approximating Steiner tree with constant ratio better than $\frac{19}{18} \approx 1.0555$ unless \textsf{P = NP}. We also relate the problem to the Unique Games Conjecture by showing that it is \textsf{UG}-hard to find a constant approximation ratio better than $\frac{17}{16} = 1.0625$. In the special case of quasi-bipartite graphs, we prove an inapproximability factor of $\frac{25}{24} \approx 1.0416$ unless \textsf{P = NP}, which improves upon the previous bound of $\frac{128}{127} \approx 1.0078$. The reductions that we present for all the cases are of the same spirit with appropriate modifications. Our main technical contribution is an adaptation of a Set-Cover type reduction in which the Long Code is used to the geometric setting of the problems we consider.