The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

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 minimum-cost 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 fixed-parameter 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 fixed-parameter tractable special cases: we show that if every pattern in H has the combinatorial property of being "transitively equivalent to a bounded-length 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 polynomial-time 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.

[1]  Anupam Gupta,et al.  Set connectivity problems in undirected graphs and the directed Steiner network problem , 2008, SODA '08.

[2]  Dániel Marx,et al.  On tree width, bramble size, and expansion , 2009, J. Comb. Theory, Ser. B.

[3]  Philip N. Klein,et al.  A Polynomial-Time Approximation Scheme for Euclidean Steiner Forest , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Michael R. Fellows,et al.  On the parameterized complexity of multiple-interval graph problems , 2009, Theor. Comput. Sci..

[5]  Jon Feldman,et al.  The Directed Steiner Network problem is tractable for a constant number of terminals , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[6]  Mohammad Taghi Hajiaghayi,et al.  Improved Approximation Algorithms for (Budgeted) Node-Weighted Steiner Problems , 2013, ICALP.

[7]  Mohammad Taghi Hajiaghayi,et al.  Approximation algorithms for node-weighted buy-at-bulk network design , 2007, SODA '07.

[8]  Pasin Manurangsi,et al.  Parameterized Approximation Algorithms for Directed Steiner Network Problems , 2017, ESA.

[9]  Rolf Niedermeier,et al.  Parameterized Complexity of Arc-Weighted Directed Steiner Problems , 2009, SIAM J. Discret. Math..

[10]  Ondrej Suchý.  On Directed Steiner Trees with Multiple Roots , 2016, WG.

[11]  Alex Zelikovsky,et al.  Tighter Bounds for Graph Steiner Tree Approximation , 2005, SIAM J. Discret. Math..

[12]  Hans L. Bodlaender,et al.  Some Classes of Graphs with Bounded Treewidth , 1988, Bull. EATCS.

[13]  Sudipto Guha,et al.  Approximation algorithms for directed Steiner problems , 1999, SODA '98.

[14]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.

[15]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[16]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[17]  Mohammad Taghi Hajiaghayi,et al.  Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth , 2009, JACM.

[18]  Jesper Nederlof.  Fast Polynomial-Space Algorithms Using Inclusion-Exclusion , 2012, Algorithmica.

[19]  Fabrizio Grandoni,et al.  Steiner Tree Approximation via Iterative Randomized Rounding , 2013, JACM.

[20]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[21]  Mohammad Taghi Hajiaghayi,et al.  Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions) , 2014, SODA.

[22]  S. E. Dreyfus,et al.  The steiner problem in graphs , 1971, Networks.

[23]  Philip N. Klein,et al.  Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs , 2009, TALG.

[24]  Alex Zelikovsky,et al.  A series of approximation algorithms for the acyclic directed steiner tree problem , 1997, Algorithmica.

[25]  Vijay V. Vazirani,et al.  On the bidirected cut relaxation for the metric Steiner tree problem , 1999, SODA '99.

[26]  Mohammad Taghi Hajiaghayi,et al.  Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[27]  Mohammad Taghi Hajiaghayi,et al.  A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands (Extended Abstract) , 2014, IPEC.

[28]  Andreas Björklund,et al.  Fourier meets möbius: fast subset convolution , 2006, STOC '07.

[29]  Mohammad Taghi Hajiaghayi,et al.  Euclidean Prize-Collecting Steiner Forest , 2011, Algorithmica.

[30]  Frank Harary,et al.  Graph Theory , 2016 .

[31]  Xinhui Wang,et al.  Dynamic Programming for Minimum Steiner Trees , 2007, Theory of Computing Systems.

[32]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[33]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[34]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[35]  Philip N. Klein,et al.  An O(n log n) approximation scheme for Steiner tree in planar graphs , 2009, TALG.