Differential approximation results for the Steiner tree problem

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−ǫ for any ǫ ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where the input graph is supposed complete and the edge distances are arbitrary, we prove that it can be di erentially approximated within 1/2. For the third one de ned on complete graphs with edge distances 1 or 2, we show that it is di erentially approximable within 0.82. Also, we extend the result of (M. Bern and P. Plassmann The Steiner problem with edge lengths 1 and 2, Inform. Process. Lett. 32, 1989), we show that the Steiner tree problem with edge lengths 1 and 2 isMaxSNP-complete even in the case where |V | 6 r|N |, for any r > 0. This allows us to nally show that Steiner tree problem with edge lengths 1 and 2 cannot by approximated by polynomial time di erential approximation schemata.