Approximation hardness of the Steiner tree problem on graphs

Steiner tree problem in weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within 96/95. Our inapproximability results are stated in parametric way and can be further improved just providing gadgets and/or expanders with better parameters. The reduction is from Hastad's inapproximability result for maximum satisfiability of linear equations modulo 2 with three unknowns per equation. This was first used for the Steiner tree problem by Thimm whose approach was the main starting point for our results.