Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parameterization. In particular, on one hand Steiner Tree is known to be ${APX}$-hard, and ${W[2]}$-hard on the other, if parameterized by the number of nonterminals (Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (${EPAS}$), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (${PSAKS}$) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For none of these is an ${EPAS}$ likely to exist for the studied parameter. For Steiner Forest an easy observation shows that the problem is ${APX}$-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is ${W[1]}$-hard. Nevertheless, we show that an ${EPAS}$ exists for Unweighted Directed Steiner Tree, but a ${PSAKS}$ does not. We also prove that there is an ${EPAS}$ and a ${PSAKS}$ for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.