Document Open Access Logo

Hitting Meets Packing: How Hard Can It Be?

Authors Jacob Focke , Fabian Frei , Shaohua Li , Dániel Marx , Philipp Schepper , Roohani Sharma , Karol Węgrzycki

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.55.pdf
  • Filesize: 1.05 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Algorithm design techniques
Keywords
  • Hitting
  • Packing
  • Covering
  • Parameterized Algorithms
  • Lower Bounds
  • Treewidth

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of 𝒳-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing 𝓁 vertex-disjoint objects of type 𝒳? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination 𝒳-HitPack can be significantly harder than these two base problems. Already for one particular choice of 𝒳, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain at the intersection of the areas of hitting and packing problems.
At a high level, we present two case studies: (1) 𝒳 being all cycles, and (2) 𝒳 being all copies of a fixed graph H. In each, we explore the classical complexity as well as the parameterized complexity with the natural parameters k+𝓁 and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph on at least 3 vertices, the problem is Σ₂^𝖯-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For 𝒳 being all cycles, we establish a 2^{poly(k+𝓁)}⋅ n^{𝒪(1)} algorithm using an involved branching method, for example. Also, for 𝒳 being all edges (i.e., H = K₂; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^{poly(tw)}⋅ n^{𝒪(1)} on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.

Cite As Get BibTex

Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki. Hitting Meets Packing: How Hard Can It Be?. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 55:1-55:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.55

Author Details

Jacob Focke
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Fabian Frei
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Shaohua Li
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Philipp Schepper
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Roohani Sharma
  • University of Bergen, Norway
Karol Węgrzycki
  • Saarland University, Saarbrücken, Germany
  • Max Planck Institute for Informatics, SIC, Saarbrücken, Germany

Funding

  • Schepper, Philipp: Part of Saarbrücken Graduate School of Computer Science, Germany.
  • Węgrzycki, Karol: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 850979).

References

  1. Isolde Adler, Martin Grohe, and Stephan Kreutzer. Computing excluded minors. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, pages 641-650. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347153.
  2. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion. ACM Trans. Algorithms, 15(1):11:1-11:28, 2019. URL: https://doi.org/10.1145/3284356.
  3. Noga Alon, Raphael Yuster, and Uri Zwick. Color-Coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  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. Stéphane Bessy, Marin Bougeret, R. Krithika, Abhishek Sahu, Saket Saurabh, Jocelyn Thiebaut, and Meirav Zehavi. Packing Arc-Disjoint Cycles in Tournaments. Algorithmica, 83(5):1393-1420, 2021. URL: https://doi.org/10.1007/s00453-020-00788-2.
  6. Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé. Kernels for feedback arc set in tournaments. J. Comput. Syst. Sci., 77(6):1071-1078, 2011. URL: https://doi.org/10.1016/j.jcss.2010.10.001.
  7. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
  8. Glencora Borradaile and Hung Le. Optimal Dynamic Program for r-Domination Problems over Tree Decompositions. In 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, volume 63 of LIPIcs, pages 8:1-8:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.8.
  9. Yixin Cao. A Naive Algorithm for Feedback Vertex Set. In 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, volume 61 of OASIcs, pages 1:1-1:9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/OASIcs.SOSA.2018.1.
  10. Yixin Cao and Dániel Marx. Chordal Editing is Fixed-Parameter Tractable. Algorithmica, 75(1):118-137, 2016. URL: https://doi.org/10.1007/s00453-015-0014-x.
  11. Maria Chudnovsky, Alexandra Ovetsky Fradkin, and Paul D. Seymour. Tournament immersion and cutwidth. J. Comb. Theory, Ser. B, 102(1):93-101, 2012. URL: https://doi.org/10.1016/j.jctb.2011.05.001.
  12. Maria Chudnovsky, Alex Scott, and Paul D. Seymour. Disjoint paths in unions of tournaments. J. Comb. Theory, Ser. B, 135:238-255, 2019. URL: https://doi.org/10.1016/j.jctb.2018.08.007.
  13. Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 1080-1099. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.70.
  14. Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 1650-1669. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch113.
  15. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. ACM Trans. Algorithms, 18(2):17:1-17:31, 2022. URL: https://doi.org/10.1145/3506707.
  16. Frank K. H. A. Dehne, Michael R. Fellows, Michael A. Langston, Frances A. Rosamond, and Kim Stevens. An 2^O(k)n³ FPT Algorithm for the Undirected Feedback Vertex Set Problem. Theory Comput. Syst., 41(3):479-492, 2007. URL: https://doi.org/10.1007/s00224-007-1345-z.
  17. 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. URL: https://doi.org/10.1016/j.jda.2009.08.001.
  18. Dorit Dor and Michael Tarsi. Graph decomposition is np-complete: A complete proof of holyer’s conjecture. SIAM J. Comput., 26(4):1166-1187, 1997. URL: https://doi.org/10.1137/S0097539792229507.
  19. László Egri, Dániel Marx, and Paweł Rzążewski. Finding List Homomorphisms from Bounded-treewidth Graphs to Reflexive Graphs: a Complete Complexity Characterization. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, volume 96 of LIPIcs, pages 27:1-27:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.27.
  20. P. Erdős and L. Pósa. On independent circuits contained in a graph. Canad. J. Math., 17:347-352, 1965. Google Scholar
  21. Michael R. Fellows and Michael A. Langston. On search, decision and the efficiency of polynomial-time algorithms (extended abstract). In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, pages 501-512. ACM, 1989. URL: https://doi.org/10.1145/73007.73055.
  22. Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki. Hitting meets packing: How hard can it be? CoRR, abs/2402.14927, 2024. URL: https://doi.org/10.48550/arXiv.2402.14927.
  23. Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, and Philip Wellnitz. Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 3664-3683. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch140.
  24. Jacob Focke, Dániel Marx, and Paweł Rzążewski. Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, pages 431-458. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.22.
  25. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL: https://doi.org/10.1145/2886094.
  26. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Hitting topological minors is FPT. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 1317-1326. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384318.
  27. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Fast Local Search Algorithm for Weighted Feedback Arc Set in Tournaments. In Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2010. AAAI Press, 2010. URL: http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/view/1829, URL: https://doi.org/10.1609/AAAI.V24I1.7557.
  28. Florent Foucaud, Esther Galby, Liana Khazaliya, Shaohua Li, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Problems in np can admit double-exponential lower bounds when parameterized by treewidth or vertex cover, 2024. To appear at ICALP 2024. URL: https://arxiv.org/abs/2307.08149.
  29. Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of stability. J. Comput. Syst. Sci., 123:103-121, 2022. URL: https://doi.org/10.1016/J.JCSS.2021.07.001.
  30. M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979. Google Scholar
  31. Jiong Guo, Jens Gramm, Falk Hüffner, Rolf Niedermeier, and Sebastian Wernicke. Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci., 72(8):1386-1396, 2006. URL: https://doi.org/10.1016/j.jcss.2006.02.001.
  32. Bart M. P. Jansen and Marcin Pilipczuk. Approximation and Kernelization for Chordal Vertex Deletion. SIAM J. Discret. Math., 32(3):2258-2301, 2018. URL: https://doi.org/10.1137/17M112035X.
  33. Naonori Kakimura, Ken-ichi Kawarabayashi, and Dániel Marx. Packing cycles through prescribed vertices. J. Comb. Theory, Ser. B, 101(5):378-381, 2011. URL: https://doi.org/10.1016/j.jctb.2011.03.004.
  34. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters, tight bounds, and approximation for (k, r)-center. Discret. Appl. Math., 264:90-117, 2019. URL: https://doi.org/10.1016/j.dam.2018.11.002.
  35. Ken-ichi Kawarabayashi and Bruce A. Reed. Odd cycle packing. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 695-704. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806785.
  36. Ken-ichi Kawarabayashi, Bruce A. Reed, and Paul Wollan. The Graph Minor Algorithm with Parity Conditions. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 27-36. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.52.
  37. David G. Kirkpatrick and Pavol Hell. On the Complexity of General Graph Factor Problems. SIAM J. Comput., 12(3):601-609, 1983. URL: https://doi.org/10.1137/0212040.
  38. Tomasz Kociumaka and Marcin Pilipczuk. Faster deterministic Feedback Vertex Set. Inf. Process. Lett., 114(10):556-560, 2014. URL: https://doi.org/10.1016/j.ipl.2014.05.001.
  39. 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, volume 47 of LIPIcs, pages 49:1-49:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.49.
  40. Michael Lampis. Model Checking Lower Bounds for Simple Graphs. Log. Methods Comput. Sci., 10(1), 2014. URL: https://doi.org/10.2168/LMCS-10(1:18)2014.
  41. Michael Lampis, Stefan Mengel, and Valia Mitsou. QBF as an alternative to courcelle’s theorem. In Theory and Applications of Satisfiability Testing - SAT 2018 - 21st International Conference, SAT 2018, Held as Part of the Federated Logic Conference, FloC 2018, volume 10929 of Lecture Notes in Computer Science, pages 235-252. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-94144-8_15.
  42. Michael Lampis and Valia Mitsou. Treewidth with a Quantifier Alternation Revisited. In 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, volume 89 of LIPIcs, pages 26:1-26:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.26.
  43. John M. Lewis and Mihalis Yannakakis. The Node-Deletion Problem for Hereditary Properties is NP-Complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  44. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
  45. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly Superexponential Parameterized Problems. SIAM J. Comput., 47(3):675-702, 2018. URL: https://doi.org/10.1137/16M1104834.
  46. Daniel Lokshtanov, Pranabendu Misra, Joydeep Mukherjee, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. 2-Approximating Feedback Vertex Set in Tournaments. ACM Trans. Algorithms, 17(2):11:1-11:14, 2021. URL: https://doi.org/10.1145/3446969.
  47. Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh, and Meirav Zehavi. Packing Cycles Faster Than Erdős Pósa. SIAM J. Discret. Math., 33(3):1194-1215, 2019. URL: https://doi.org/10.1137/17M1150037.
  48. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster Parameterized Algorithms Using Linear Programming. ACM Trans. Algorithms, 11(2):15:1-15:31, 2014. URL: https://doi.org/10.1145/2566616.
  49. Dániel Marx. Chordal Deletion is Fixed-Parameter Tractable. Algorithmica, 57(4):747-768, 2010. URL: https://doi.org/10.1007/s00453-008-9233-8.
  50. Dániel Marx. Chordless Cycle Packing Is Fixed-Parameter Tractable. In 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, volume 173 of LIPIcs, pages 71:1-71:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.71.
  51. Dániel Marx and Valia Mitsou. Double-Exponential and Triple-Exponential Bounds for Choosability Problems Parameterized by Treewidth. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 28:1-28:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.28.
  52. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, volume 198 of LIPIcs, pages 95:1-95:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.95.
  53. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard). In 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, volume 249 of LIPIcs, pages 22:1-22:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.22.
  54. Pranabendu Misra, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. A Polynomial Kernel for Feedback Arc Set on Bipartite Tournaments. Theory Comput. Syst., 53(4):609-620, 2013. URL: https://doi.org/10.1007/s00224-013-9453-4.
  55. Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs. In 28th Annual European Symposium on Algorithms, ESA 2020, volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.74.
  56. Karolina Okrasa and Paweł Rzążewski. Fine-Grained Complexity of the Graph Homomorphism Problem for Bounded-Treewidth Graphs. SIAM J. Comput., 50(2):487-508, 2021. URL: https://doi.org/10.1137/20M1320146.
  57. Guoqiang Pan and Moshe Y. Vardi. Fixed-Parameter Hierarchies inside PSPACE. In 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings, pages 27-36. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/LICS.2006.25.
  58. M. Pontecorvi and Paul Wollan. Disjoint cycles intersecting a set of vertices. J. Comb. Theory, Ser. B, 102(5):1134-1141, 2012. URL: https://doi.org/10.1016/j.jctb.2012.05.004.
  59. Venkatesh Raman and Saket Saurabh. Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci., 351(3):446-458, 2006. URL: https://doi.org/10.1016/j.tcs.2005.10.010.
  60. M. S. Ramanujan and Saket Saurabh. Linear-Time Parameterized Algorithms via Skew-Symmetric Multicuts. ACM Trans. Algorithms, 13(4):46:1-46:25, 2017. URL: https://doi.org/10.1145/3128600.
  61. Bruce A. Reed. Mangoes and Blueberries. Combinatorica, 19(2):267-296, 1999. URL: https://doi.org/10.1007/s004930050056.
  62. Bruce A. Reed, Neil Robertson, Paul D. Seymour, and Robin Thomas. Packing Directed Circuits. Combinatorica, 16(4):535-554, 1996. URL: https://doi.org/10.1007/BF01271272.
  63. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Oper. Res. Lett., 32(4):299-301, 2004. URL: https://doi.org/10.1016/j.orl.2003.10.009.
  64. Marcus Schaefer and Christopher Umans. SIGACT news complexity theory column 37. Guest column: Completeness in the polynomial-time hierarchy: Part I: A compendium. SIGACT News, 33(3):32-49, 2002. URL: https://doi.org/10.1145/582475.582484.
  65. S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. Google Scholar
  66. Christopher Umans. Hardness of Approximating Sigma2P Minimization Problems. In 40th Annual Symposium on Foundations of Computer Science, FOCS 99, pages 465-474. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814619.
  67. Magnus Wahlström. Representative set statements for delta-matroids and the mader delta-matroid. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 780-810. SIAM, 2024. Google Scholar
  68. Mingyu Xiao and Jiong Guo. A Quadratic Vertex Kernel for Feedback Arc Set in Bipartite Tournaments. Algorithmica, 71(1):87-97, 2015. URL: https://doi.org/10.1007/s00453-013-9783-2.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact