STAMM@DBNINF5.BITNET (Hermann Stamm) (10/04/90)
I am interested in the following problem ( which this time is truely NP-complete ! ) : ------------------------------------------------------------------------------ MINIMUM-DIRECTED-STEINER-TREE (MDStT) Instance: Strongly connected digraph D=(V,A), vertex s \in V, T \subset V, K < |V| Question: Does there exist a directed Steiner tree S=(V',A') from s into T with |A'| < K ? * ( a subtree S of D, so that s--->t for all t \in T ) ------------------------------------------------------------------------------ I have found only two references which handle the above problem, [NAST74] and [WONG84]. In [WONG84] it is shown, that MDStT is NP-complete. For the undirected Steiner tree problem there are approximation algorithms with a worst-case-ratio of 2. My question now is: Is there any approximation algorithm with constant ( or at least logarithmic ) worst-case-ratio bound, or, if not, at least in the case of a planar digraph D ? ( The idea of a Travelling-Salesman tour after the doubling of a minimum spanning tree in the proof of the undirected worst-case-ratio bound 2 does not work in the directed case ! ) Thanks in advance, Hermann Stamm Universitaet Bonn STAMM@DBNINF5.BITNET ------------------------------------------------------------------------------ [NAST74] Nastansky, L. , Selkow, S.M. , Stewart, N.F. , Zeitschrift fuer Operations Research , 1974 , Vol. 18 , pp. 59-67 , Cost-Minimal Trees in Directed Acyclic Graphs [WONG84] Wong, R.T. , Mathematical Programming , 1984 , Vol. 28 , pp. 271-287 , A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph ------------------------------------------------------------------------------