Abstract
Given a directed graph G and a list (s_1, t_1), ..., (s_k, t_k) of terminal pairs, the Directed Steiner Network problem asks for a minimumcost subgraph of G that contains a directed s_i > t_i path for every 1 <= i <= k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t_1, . . . , t_k) is known to be fixedparameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t_i to every other t_j ) is known to be W[1]hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]hard. Formally, if H is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list (s_1, t_1), ..., (s_k, t_k) of requests form a directed graph that is a member of H. Our main result is a complete characterization of the classes H resulting in fixedparameter tractable special cases: we show that if every pattern in H has the combinatorial property of being "transitively equivalent to a boundedlength caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]hard for every recursively enumerable H not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], Strongly Connected Steiner Subgraph is W[1]hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomialtime for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.
BibTeX  Entry
@InProceedings{feldmann_et_al:LIPIcs:2016:6306,
author = {Andreas Emil Feldmann and D{\'a}niel Marx},
title = {{The Complexity Landscape of FixedParameter Directed Steiner Network Problems}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {27:127:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770132},
ISSN = {18688969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6306},
URN = {urn:nbn:de:0030drops63060},
doi = {10.4230/LIPIcs.ICALP.2016.27},
annote = {Keywords: Directed Steiner Tree, Directed Steiner Network, fixedparameter tractability, dichotomy}
}
Keywords: 

Directed Steiner Tree, Directed Steiner Network, fixedparameter tractability, dichotomy 
Seminar: 

43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) 
Issue Date: 

2016 
Date of publication: 

17.08.2016 