On the approximability of Dense Steiner Problems

Abstract

We study the approximation complexity of the @e-Dense Steiner Tree Problem which was introduced by Karpinski and Zelikovsky (1998) [13]. They proved that for each @e>0, this problem admits a PTAS. Based on their method we consider here dense versions of various Steiner Tree problems. In particular, we give polynomial time approximation schemes for the @e-Dense k-Steiner Tree Problem, the @e-Dense Prize Collecting Steiner Tree Problem and the @e-Dense Group Steiner Tree Problem. We also show that the @e-Dense Steiner Forest Problem is approximable within ratio 1+O((@?ilog|Si|)/(@?i|Si|)) where S1,...,Sn are the terminal sets of the given instance. This ratio becomes small when the number of terminal sets is small compared to the total number of terminals.