Sub-Exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number
Fradkin and Seymour [Journal of Combinatorial Graph Theory, Series B, 2015] defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk [ESA, 2013], where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is that the yes-instances of the problems above have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, an inductive argument and structural properties of the digraphs.
sub-exponential fixed-parameter tractable algorithms
directed feedback arc set
directed cutwidth
optimal linear arrangement
bounded independence number digraph
Theory of computation~Fixed parameter tractability
Mathematics of computing~Combinatorial algorithms
35:1-35:19
Regular Paper
Pranabendu
Misra
Pranabendu Misra
University of Bergen, Norway
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI, India
Roohani
Sharma
Roohani Sharma
Institute of Mathematical Sciences, HBNI, India
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University, Beersheba, Israel
10.4230/LIPIcs.FSTTCS.2018.35
Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: Ranking and clustering. J. ACM, 55(5):23:1-23:27, 2008.
Noga Alon. Ranking Tournaments. SIAM J. Discrete Math., 20(1):137-142, 2006.
Noga Alon, Daniel Lokshtanov, and Saket Saurabh. Fast FAST. In Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, pages 49-58, 2009.
Jørgen Bang-Jensen and Carsten Thomassen. A Polynomial Algorithm for the 2-Path Problem for Semicomplete Digraphs. SIAM J. Discrete Math., 5(3):366-376, 1992.
Florian Barbero, Christophe Paul, and Michal Pilipczuk. Exploring the Complexity of Layout Parameters in Tournaments and Semi-Complete Digraphs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 70:1-70:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments. Combinatorics, Probability & Computing, 16(1):1-4, 2007.
Maria Chudnovsky, Alexandra Ovetsky Fradkin, and Paul D. Seymour. Tournament immersion and cutwidth. J. Comb. Theory, Ser. B, 102(1):93-101, 2012.
Maria Chudnovsky and Paul D. Seymour. A well-quasi-order for tournaments. J. Comb. Theory, Ser. B, 101(1):47-53, 2011.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Michael Dom, Jiong Guo, Falk Hüffner, Rolf Niedermeier, and Anke Truß. Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms, 8(1):76-86, 2010.
Uriel Feige. Faster FAST(Feedback Arc Set in Tournaments). CoRR, abs/0911.5094, 2009. URL: http://arxiv.org/abs/0911.5094.
http://arxiv.org/abs/0911.5094
Fedor V. Fomin and Michal Pilipczuk. Jungles, bundles, and fixed parameter tractability. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 396-413, 2013.
Fedor V. Fomin and Michal Pilipczuk. Subexponential Parameterized Algorithm for Computing the Cutwidth of a Semi-complete Digraph. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 505-516, 2013.
Alexandra Ovetsky Fradkin and Paul D. Seymour. Tournament pathwidth and topological containment. J. Comb. Theory, Ser. B, 103(3):374-384, 2013.
Alexandra Ovetsky Fradkin and Paul D. Seymour. Edge-disjoint paths in digraphs with bounded independence number. J. Comb. Theory, Ser. B, 110:19-46, 2015.
T Gallai and AN Milgram. Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sc. Math, 21:181-186, 1960.
Marek Karpinski and Warren Schudy. Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament. In Algorithms and Computation - 21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15-17, 2010, Proceedings, Part I, pages 3-14, 2010.
Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 95-103, 2007.
Mithilesh Kumar and Daniel Lokshtanov. Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, volume 47 of LIPIcs, pages 49:1-49:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016.
Matthias Mnich, Virginia Vassilevska Williams, and László A. Végh. A 7/3-Approximation for Feedback Vertex Sets in Tournaments. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 67:1-67:14, 2016.
Michal Pilipczuk. Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 197-208. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013.
Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi
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