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A 9/4-Approximation for Directed Feedback Vertex Sets in Quasi-Transitive Digraphs

Authors Ebrahim Ghorbani , Matthias Mnich

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LIPIcs.ICALP.2026.96.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • directed feedback vertex set
  • tournaments
  • quasi-transitive digraphs

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Abstract

We provide the first non-trivial approximation algorithm for the fundamental directed feedback vertex set (DFVS) problem in the class of quasi-transitive digraphs. This class of digraphs encompasses both dense and sparse classes of digraphs, for which specialized DFVS algorithms were proposed in the literature, like tournaments or transitive orientations of bounded treewidth graphs.
Our approximation algorithm can handle both dense graphs, as well as sparse graphs, by a single approach, which is based on carefully analysing the solutions to a linear programming relaxation of DFVS. It also handles the node-weighted DFVS problem, for which it computes a 9/4-approximation in polynomial time.
Along the way, we improve and simplify the best-known deterministic polynomial-time approximation algorithms for DFVS in tournaments (Cai et al., SICOMP 2001; Mnich et al., ESA 2016).

Cite As Get BibTex

Ebrahim Ghorbani and Matthias Mnich. A 9/4-Approximation for Directed Feedback Vertex Sets in Quasi-Transitive Digraphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 96:1-96:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.96

Author Details

Ebrahim Ghorbani
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Germany
Matthias Mnich
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Germany

References

  1. Manuel Aprile, Matthew Drescher, Samuel Fiorini, and Tony Huynh. A 7/3-approximation algorithm for feedback vertex set in tournaments via Sherali–Adams. Discrete Appl. Math., 337:149-160, 2023. URL: https://doi.org/10.1016/J.DAM.2023.04.016.
  2. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math., 12(3):289-297, 1999. URL: https://doi.org/10.1137/S0895480196305124.
  3. Jørgen Bang-Jensen and Gregory Gutin. Classes of directed graphs, volume 11. Springer, 2018. Google Scholar
  4. Jørgen Bang-Jensen, Alessandro Maddaloni, and Saket Saurabh. Algorithms and kernels for feedback set problems in generalizations of tournaments. Algorithmica, 76(2):320-343, 2016. URL: https://doi.org/10.1007/S00453-015-0038-2.
  5. Jørgen Bang-Jensen and Jing Huang. Quasi-transitive digraphs. J. Graph Theory, 20(2):141-161, 1995. URL: https://doi.org/10.1002/JGT.3190200205.
  6. Ann Becker and Dan Geiger. Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell., 83(1):167-188, 1996. Google Scholar
  7. Piotr Berman and Grigory Yaroslavtsev. Primal-dual approximation algorithms for node-weighted network design in planar graphs. In Proc. APPROX 2012, volume 7408 of Lecture Notes Comput. Sci., pages 50-60, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_5.
  8. Marthe Bonamy, Łukasz Kowalik, Jesper Nederlof, Michał Pilipczuk, Arkadiusz Socała, and Marcin Wrochna. On directed feedback vertex set parameterized by treewidth. In Proc. WG 2018, volume 11159 of Lecture Notes Comput. Sci., pages 65-78, 2018. URL: https://doi.org/10.1007/978-3-030-00256-5_6.
  9. Mao-Cheng Cai, Xiaotie Deng, and Wenan Zang. An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput., 30(6):1993-2007, 2001. URL: https://doi.org/10.1137/S0097539798338163.
  10. Jason Cong. Computing maximum weighted k-families and k-cofamilies in partially ordered sets. Technical Report CSD-930014, University of California, 1993. URL: https://ftp.cs.ucla.edu/tech-report/1993-reports/930014.pdf.
  11. Guy Even, Joseph Naor, Baruch Schieber, and Madhu Sudan. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20(2):151-174, 1998. URL: https://doi.org/10.1007/PL00009191.
  12. Michel X Goemans and David P Williamson. Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica, 18(1):37-59, 1998. URL: https://doi.org/10.1007/PL00009810.
  13. Curtis Greene and Daniel J. Kleitman. The structure of Sperner k-families. J. Combinatorial Theory Ser. A, 20(1):41-68, 1976. URL: https://doi.org/10.1016/0097-3165(76)90077-7.
  14. Sushmita Gupta, Sounak Modak, Saket Saurabh, and Sanjay Seetharaman. Quick-sort style approximation algorithms for generalizations of feedback vertex set in tournaments. In Proc. LATIN 2024, volume 14578 of Lecture Notes Comput. Sci., pages 225-240, 2024. URL: https://doi.org/10.1007/978-3-031-55598-5_15.
  15. Venkatesan Guruswami and Euiwoong Lee. Simple proof of hardness of feedback vertex set. Theory Comput., 12(6):1-11, 2016. URL: https://doi.org/10.4086/TOC.2016.V012A006.
  16. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  17. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335-349, 2008. Google Scholar
  18. Daniel Lokshtanov, Pranabendu Misra, Joydeep Mukherjee, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. 2-approximating feedback vertex set in tournaments. In Proc. SODA 2020, pages 1010-1018, 2020. URL: https://doi.org/10.1137/1.9781611975994.61.
  19. L. Lovász. Normal hypergraphs and the perfect graph conjecture. Discrete Math., 2(3):253-267, 1972. URL: https://doi.org/10.1016/0012-365X(72)90006-4.
  20. L. Mirsky. A dual of Dilworth’s decomposition theorem. Amer. Math. Monthly, 78:876-877, 1971. Google Scholar
  21. Matthias Mnich, Virginia Vassilevska Williams, and László A. Végh. A 7/3-approximation for feedback vertex sets in tournaments. In Proc. ESA 2016, volume 57 of Leibniz Int. Proc. Informatics, pages 67:1-67:14, 2016. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.67.
  22. M. Sarrafzadeh and R.-D. Lou. Maximum k-covering of weighted transitive graphs with applications. Algorithmica, 9(1):84-100, 1993. URL: https://doi.org/10.1007/BF01185340.
  23. Paul D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995. URL: https://doi.org/10.1007/BF01200760.
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