A Polynomial - Time Subpolynom - Approximation Scheme for

The acyclic directed Steiner tree problem (ADSP) requires a minimal outward tree within an acyclic digraph with edge costs G = (V; E; d) which connects a root r with a distinguished subset S V , #S = k. The best possible performance guarantee of any polynomial approximation algorithm for ADSP cannot be less than 1 4 log k unless ~ P NP. The presented series of heuristics A n has a performance guarantee k 1 n (1 + ln k) n?1. This implies that that fA n g is a polynomial exp p 4 ln k ln(ln k + 1)?ln(ln k +1)]-approximation scheme for ADSP. Research partially supprted by the Volkswagen{Stiftung.