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The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

Published: 12 December 2023 Publication History

Abstract

Given a directed graph G and a list (s1, t1), …, (sd, td) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed siti path for every 1≤ id. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, …, td) 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 ti to every other tj) 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 (s1, t1), …, (sd, td) of demands 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], q-Root Steiner Tree is FPT for constant q [Suchý, WG 2016], 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.

References

[1]
Ajit Agrawal, Philip N. Klein, and R. Ravi. 1995. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24, 3 (1995), 440–456.
[2]
Aaron Archer, MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Howard J. Karloff. 2011. Improved approximation algorithms for prize-collecting Steiner tree and TSP. SIAM J. Comput. 40, 2 (2011), 309–332.
[3]
MohammadHossein Bateni and Mohammad Taghi Hajiaghayi. 2012. Euclidean prize-collecting Steiner forest. Algorithmica 62, 3-4 (2012), 906–929.
[4]
MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Vahid Liaghat. 2013. Improved approximation algorithms for (budgeted) node-weighted Steiner problems. In 40th International Colloquium on Automata, Languages, and Programming. 81–92.
[5]
MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Dániel Marx. 2011. Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM 58, 5 (2011), 21.
[6]
Hans L. Bodlaender. 1988. Some classes of graphs with bounded treewidth. Bulletin of the EATCS 36 (1988), 116–125.
[7]
Glencora Borradaile, Philip N. Klein, and Claire Mathieu. 2009. An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms 5, 3 (2009).
[8]
Glencora Borradaile, Philip N. Klein, and Claire Mathieu. 2015. A polynomial-time approximation scheme for euclidean Steiner forest. ACM Transactions on Algorithms 11, 3 (2015), 19:1–19:20.
[9]
Jaroslaw Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. 2013. Steiner tree approximation via iterative randomized rounding. J. ACM 60, 1 (2013), 6.
[10]
Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. 1999. Approximation algorithms for directed Steiner problems. J. Algorithms 33, 1 (1999), 73–91.
[11]
Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. 2011. Set connectivity problems in undirected graphs and the directed Steiner network problem. ACM Transactions on Algorithms 7, 2 (2011), 18.
[12]
Chandra Chekuri, Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R. Salavatipour. 2007. Approximation algorithms for node-weighted buy-at-bulk network design. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms. 1265–1274. http://dl.acm.org/citation.cfm?id=1283383.1283519.
[13]
Rajesh Chitnis, Andreas Emil Feldmann, and Pasin Manurangsi. 2018. Parameterized approximation algorithms for bidirected Steiner network problems. In 26th Annual European Symposium on Algorithms, ESA. 20:1–20:16.
[14]
Rajesh Hemant Chitnis, Hossein Esfandiari, Mohammad Taghi Hajiaghayi, Rohit Khandekar, Guy Kortsarz, and Saeed Seddighin. 2014. A tight algorithm for strongly connected Steiner subgraph on two terminals with demands (extended abstract). In 9th International Symposium on Parameterized and Exact Computation. 159–171.
[15]
Rajesh Hemant Chitnis, Andreas Emil Feldmann, Mohammad Taghi Hajiaghayi, and Dániel Marx. 2020. Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions). SIAM J. Comput. 49, 2 (2020), 318–364.
[16]
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. 2015. Parameterized Algorithms. Springer.
[17]
Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Philip N. Klein. 2014. Node-weighted Steiner tree and group Steiner tree in planar graphs. ACM Transactions on Algorithms 10, 3 (2014), 13:1–13:20.
[18]
Reinhard Diestel. 2005. Graph Theory (3rd ed.). Graduate Texts in Mathematics, Vol. 173. Springer-Verlag, Berlin. xvi+411 pages.
[19]
R. G. Downey and M. R. Fellows. 2013. Fundamentals of Parameterized Complexity. Vol. 4. Springer.
[20]
S. E. Dreyfus and R. A. Wagner. 1971. The Steiner problem in graphs. Networks 1, 3 (1971), 195–207.
[21]
Eduard Eiben, Dusan Knop, Fahad Panolan, and Ondrej Suchý. 2019. Complexity of the Steiner network problem with respect to the number of terminals. In STACS. 25:1–25:17.
[22]
Jon Feldman and Matthias Ruhl. 2006. The directed Steiner network problem is tractable for a constant number of terminals. SIAM J. Comput. 36, 2 (2006), 543–561.
[23]
Andreas Emil Feldmann and Dániel Marx. 2016. The complexity landscape of fixed-parameter directed Steiner network problems. In 43rd International Colloquium on Automata, Languages, and Programming, (ICALP). 27:1–27:14.
[24]
Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. 2009. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410, 1 (2009), 53–61.
[25]
Jörg Flum and Martin Grohe. 2006. Parameterized Complexity Theory. Springer.
[26]
Bernhard Fuchs, Walter Kern, D. Molle, Stefan Richter, Peter Rossmanith, and Xinhui Wang. 2007. Dynamic programming for minimum Steiner trees. Theory of Computing Systems 41, 3 (2007), 493–500.
[27]
Martin Grohe and Dániel Marx. 2009. On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99, 1 (2009), 218–228.
[28]
Jiong Guo, Rolf Niedermeier, and Ondrej Suchý. 2011. Parameterized complexity of arc-weighted directed Steiner problems. SIAM J. Discrete Math. 25, 2 (2011), 583–599.
[29]
Richard M. Karp. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations. Plenum, 85–103.
[30]
Philip N. Klein and R. Ravi. 1995. A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19, 1 (1995), 104–115.
[31]
Jesper Nederlof. 2013. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica 65, 4 (2013), 868–884.
[32]
Sridhar Rajagopalan and Vijay V. Vazirani. 1999. On the bidirected cut relaxation for the metric Steiner tree problem. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms. 742–751. http://dl.acm.org/citation.cfm?id=314500.314909.
[33]
Gabriel Robins and Alexander Zelikovsky. 2005. Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19, 1 (2005), 122–134.
[34]
Ondřej Suchý. 2016. On directed Steiner trees with multiple roots. In International Workshop on Graph-Theoretic Concepts in Computer Science (WG). 257–268.
[35]
Alexander Zelikovsky. 1997. A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica 18, 1 (1997), 99–110.

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Published In

cover image ACM Transactions on Computation Theory
ACM Transactions on Computation Theory  Volume 15, Issue 3-4
December 2023
105 pages
ISSN:1942-3454
EISSN:1942-3462
DOI:10.1145/3637091
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 12 December 2023
Online AM: 06 June 2023
Accepted: 12 January 2023
Received: 15 September 2020
Published in TOCT Volume 15, Issue 3-4

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  1. Directed Steiner networks
  2. fixed-parameter tractability

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  • ERC
  • ERC Consolidator Grant SYSTEMATICGRAPH
  • Czech Science Foundation GAČR

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  • (2024)Fixed parameter tractable algorithms for watchman route related problems on line segment arrangementsInternational Journal of Computer Mathematics: Computer Systems Theory10.1080/23799927.2024.23573239:3(131-138)Online publication date: 27-May-2024

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