Document Open Access Logo

Unified Almost Linear Kernels for Generalized Covering and Packing Problems on Nowhere Dense Classes

Authors Jungho Ahn , Jinha Kim , O-joung Kwon



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2023.5.pdf
  • Filesize: 0.82 MB
  • 19 pages

Document Identifiers

Author Details

Jungho Ahn
  • Korea Institute for Advanced Study, Seoul, South Korea
Jinha Kim
  • Department of Mathematics, Chonnam National University, Gwangju, South Korea
O-joung Kwon
  • Department of Mathematics, Hanyang University, Seoul, South Korea
  • Discrete Mathematics Group, Institute for Basic Science, Daejeon, South Korea

Cite AsGet BibTex

Jungho Ahn, Jinha Kim, and O-joung Kwon. Unified Almost Linear Kernels for Generalized Covering and Packing Problems on Nowhere Dense Classes. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 5:1-5:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.5

Abstract

Let ℱ be a family of graphs, and let p,r be nonnegative integers. For a graph G and an integer k, the (p,r,ℱ)-Covering problem asks whether there is a set D ⊆ V(G) of size at most k such that if the p-th power of G has an induced subgraph isomorphic to a graph in ℱ, then it is at distance at most r from D. The (p,r,ℱ)-Packing problem asks whether G^p has k induced subgraphs H₁,…,H_k such that each H_i is isomorphic to a graph in ℱ, and for i,j ∈ {1,…,k}, the distance between V(H_i) and V(H_j) in G is larger than r. We show that for every fixed nonnegative integers p,r and every fixed nonempty finite family ℱ of connected graphs, (p,r,ℱ)-Covering with p ≤ 2r+1 and (p,r,ℱ)-Packing with p ≤ 2⌊r/2⌋+1 admit almost linear kernels on every nowhere dense class of graphs, parameterized by the solution size k. As corollaries, we prove that Distance-r Vertex Cover, Distance-r Matching, ℱ-Free Vertex Deletion, and Induced-ℱ-Packing for any fixed finite family ℱ of connected graphs admit almost linear kernels on every nowhere dense class of graphs. Our results extend the results for Distance-r Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and for Distance-r Independent Set by Pilipczuk and Siebertz (EJC 2021).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • kernelization
  • independent set
  • dominating set
  • covering
  • packing

Metrics

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

References

  1. Faisal N. Abu-Khzam. A kernelization algorithm for d-hitting set. J. Comput. System Sci., 76(7):524-531, 2010. URL: https://doi.org/10.1016/j.jcss.2009.09.002.
  2. Hans Adler and Isolde Adler. Interpreting nowhere dense graph classes as a classical notion of model theory. European J. Combin., 36:322-330, 2014. URL: https://doi.org/10.1016/j.ejc.2013.06.048.
  3. Jochen Alber, Michael R. Fellows, and Rolf Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363-384, 2004. URL: https://doi.org/10.1145/990308.990309.
  4. Noga Alon and Shai Gutner. Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica, 54(4):544-556, 2009. URL: https://doi.org/10.1007/s00453-008-9204-0.
  5. Manuel Aprile, Matthew Drescher, Samuel Fiorini, and Tony Huynh. A tight approximation algorithm for the cluster vertex deletion problem. Math. Program., 197(2):1069-1091, 2023. URL: https://doi.org/10.1007/s10107-021-01744-w.
  6. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) kernelization. J. ACM, 63(5):Art. 44, 69, 2016. URL: https://doi.org/10.1145/2973749.
  7. Flavia Bonomo-Braberman, Julliano R. Nascimento, Fabiano S. Oliveira, Uéverton S. Souza, and Jayme L. Szwarcfiter. Linear-time algorithms for eliminating claws in graphs. In Computing and combinatorics, volume 12273 of Lecture Notes in Comput. Sci., pages 14-26. Springer, Cham, 2020. Google Scholar
  8. H. Brönnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom., 14(4):463-479, 1995. ACM Symposium on Computational Geometry. URL: https://doi.org/10.1007/BF02570718.
  9. Santiago Canales, Gregorio Hernández, Mafalda Martins, and Inês Matos. Distance domination, guarding and covering of maximal outerplanar graphs. Discrete Appl. Math., 181:41-49, 2015. URL: https://doi.org/10.1016/j.dam.2014.08.040.
  10. Holger Dell and Dániel Marx. Kernelization of packing problems. In 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 68-81. ACM, New York, 2012. Google Scholar
  11. Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005. URL: https://doi.org/10.1145/1101821.1101823.
  12. Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer, Berlin, fifth edition, 2018. Paperback edition of [MR3644391]. Google Scholar
  13. Frederic Dorn. Dynamic programming and fast matrix multiplication. In 14th Annual European Symposium, Zurich, Switzerland, September 11-13, 2006, volume 4168 of Lecture Notes in Comput. Sci., pages 280-291. Springer, Berlin, 2006. URL: https://doi.org/10.1007/11841036_27.
  14. Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness. I. Basic results. SIAM J. Comput., 24(4):873-921, 1995. URL: https://doi.org/10.1137/S0097539792228228.
  15. Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness. II. On completeness for W[1]. Theoret. Comput. Sci., 141(1-2):109-131, 1995. URL: https://doi.org/10.1016/0304-3975(94)00097-3.
  16. Rodney G. Downey and Michael R. Fellows. Fundamentals of parameterized complexity. Texts in Computer Science. Springer, London, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  17. Pål Grønås Drange, Markus Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and sparseness: the case of dominating set. In 33rd Symposium on Theoretical Aspects of Computer Science, volume 47 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 31, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.31.
  18. Zdeněk Dvořák. Constant-factor approximation of the domination number in sparse graphs. European J. Combin., 34(5):833-840, 2013. URL: https://doi.org/10.1016/j.ejc.2012.12.004.
  19. Jack Edmonds. Paths, trees, and flowers. Canadian J. Math., 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  20. Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michał Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood complexity and kernelization for nowhere dense classes of graphs. In 44th International Colloquium on Automata, Languages, and Programming, volume 80 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 63, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.63.
  21. Guy Even, Dror Rawitz, and Shimon Shahar. Hitting sets when the VC-dimension is small. Inform. Process. Lett., 95(2):358-362, 2005. URL: https://doi.org/10.1016/j.ipl.2005.03.010.
  22. Grzegorz Fabiański, Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. Progressive algorithms for domination and independence. In 36th International Symposium on Theoretical Aspects of Computer Science, volume 126 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 27, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. Google Scholar
  23. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Linear kernels for (connected) dominating set on H-minor-free graphs. In 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 82-92. ACM, New York, 2012. Google Scholar
  24. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Kernels for (connected) dominating set on graphs with excluded topological minors. ACM Trans. Algorithms, 14(1):Art. 6, 31, 2018. URL: https://doi.org/10.1145/3155298.
  25. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. SIAM J. Comput., 49(6):1397-1422, 2020. URL: https://doi.org/10.1137/16M1080264.
  26. Fedor V. Fomin and Dimitrios M. Thilikos. Fast parameterized algorithms for graphs on surfaces: linear kernel and exponential speed-up. In Automata, languages and programming, volume 3142 of Lecture Notes in Comput. Sci., pages 581-592. Springer, Berlin, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_50.
  27. Fedor V. Fomin and Dimitrios M. Thilikos. Dominating sets in planar graphs: branch-width and exponential speed-up. SIAM J. Comput., 36(2):281-309, 2006. URL: https://doi.org/10.1137/S0097539702419649.
  28. Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. J. Comput. System Sci., 84:219-242, 2017. URL: https://doi.org/10.1016/j.jcss.2016.09.002.
  29. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3):Art. 17, 32, 2017. URL: https://doi.org/10.1145/3051095.
  30. Shai Gutner. Polynomial kernels and faster algorithms for the dominating set problem on graphs with an excluded minor. In Parameterized and exact computation, volume 5917 of Lecture Notes in Comput. Sci., pages 246-257. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-11269-0_20.
  31. Iyad Kanj, Michael J. Pelsmajer, Marcus Schaefer, and Ge Xia. On the induced matching problem. J. Comput. System Sci., 77(6):1058-1070, 2011. URL: https://doi.org/10.1016/j.jcss.2010.09.001.
  32. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Trans. Algorithms, 12(2):Art. 21, 41, 2016. URL: https://doi.org/10.1145/2797140.
  33. Stephan Kreutzer, Roman Rabinovich, and Sebastian Siebertz. Polynomial kernels and wideness properties of nowhere dense graph classes. ACM Trans. Algorithms, 15(2):Art. 24, 19, 2019. URL: https://doi.org/10.1145/3274652.
  34. Jiří Matoušek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. URL: https://doi.org/10.1007/978-1-4613-0039-7.
  35. Hannes Moser and Somnath Sikdar. The parameterized complexity of the induced matching problem. Discrete Appl. Math., 157(4):715-727, 2009. URL: https://doi.org/10.1016/j.dam.2008.07.011.
  36. James Nastos and Yong Gao. Bounded search tree algorithms for parametrized cograph deletion: efficient branching rules by exploiting structures of special graph classes. Discrete Math. Algorithms Appl., 4(1):1250008, 23, 2012. URL: https://doi.org/10.1142/S1793830912500085.
  37. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion. I. Decompositions. European J. Combin., 29(3):760-776, 2008. URL: https://doi.org/10.1016/j.ejc.2006.07.013.
  38. Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond. ACM Trans. Algorithms, 9(1):Art. 11, 23, 2012. URL: https://doi.org/10.1145/2390176.2390187.
  39. Michał Pilipczuk and Sebastian Siebertz. Kernelization and approximation of distance-r independent sets on nowhere dense graphs. European J. Combin., 94:103309, 19, 2021. URL: https://doi.org/10.1016/j.ejc.2021.103309.
  40. J. A. Telle and Y. Villanger. FPT algorithms for domination in sparse graphs and beyond. Theoret. Comput. Sci., 770:62-68, 2019. URL: https://doi.org/10.1016/j.tcs.2018.10.030.
  41. Dekel Tsur. Faster parameterized algorithm for cluster vertex deletion. Theory Comput. Syst., 65(2):323-343, 2021. URL: https://doi.org/10.1007/s00224-020-10005-w.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail