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Solving Partial Dominating Set and Related Problems Using Twin-Width

Authors Jakub Balabán , Daniel Mock , Peter Rossmanith

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Partial Dominating Set
  • Partial Vertex Cover
  • meta-algorithm
  • counting logic
  • twin-width

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Abstract

Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are W[1]-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form ϕ≡∃ x₁⋯ ∃ x_k ∑_{α ∈ I} #y ψ_α(x₁,…,x_k,y) ≥ t, where ψ_α is a quantifier-free formula for each α ∈ I, t is an arbitrary number, and #y is a counting quantifier, can be evaluated in time f(d,k)n, where n is the number of vertices and d is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.

Cite As Get BibTex

Jakub Balabán, Daniel Mock, and Peter Rossmanith. Solving Partial Dominating Set and Related Problems Using Twin-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.13

Author Details

Jakub Balabán
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Daniel Mock
  • Dept. of Computer Science, RWTH Aachen University, Germany
Peter Rossmanith
  • Dept. of Computer Science, RWTH Aachen University, Germany

Funding

Daniel Mock and Peter Rossmanith: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Science Foundation) - DFG-927/15-2
  • Balabán, Jakub: Brno Ph.D. Talent Scholarship Holder – Funded by the Brno City Municipality.

Acknowledgements

We would like to thank one of the anonymous reviewers for suggesting a generalization of the logic we considered that also captures the Partial Vertex Cover problem.

References

  1. Jochen Alber, Hans L. Bodlaender, Henning Fernau, Ton Kloks, and Rolf Niedermeier. Fixed parameter algorithms for DOMINATING SET and related problems on planar graphs. Algorithmica, 33(4):461-493, 2002. URL: https://doi.org/10.1007/S00453-001-0116-5.
  2. Hagit Attiya and Jennifer L. Welch. Distributed computing - fundamentals, simulations, and advanced topics (2. ed.). Wiley series on parallel and distributed computing. Wiley, 2004. Google Scholar
  3. Jakub Balabán, Daniel Mock, and Peter Rossmanith. Solving partial dominating set and related problems using twin-width. CoRR, abs/2504.18218, 2025. URL: https://doi.org/10.48550/arXiv.2504.18218.
  4. R. Balasubramanian, Michael R. Fellows, and Venkatesh Raman. An improved fixed-parameter algorithm for vertex cover. Inf. Process. Lett., 65(3):163-168, 1998. URL: https://doi.org/10.1016/S0020-0190(97)00213-5.
  5. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: max independent set, min dominating set, and coloring. SIAM J. Comput., 53(5):1602-1640, 2024. URL: https://doi.org/10.1137/21M142188X.
  6. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 601-612. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00062.
  7. L. Sunil Chandran and Fabrizio Grandoni. Refined memorization for vertex cover. Inf. Process. Lett., 93(3):123-131, 2005. URL: https://doi.org/10.1016/J.IPL.2004.10.003.
  8. Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex cover: Further observations and further improvements. J. Algorithms, 41(2):280-301, 2001. URL: https://doi.org/10.1006/JAGM.2001.1186.
  9. Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theor. Comput. Sci., 411(40-42):3736-3756, 2010. URL: https://doi.org/10.1016/J.TCS.2010.06.026.
  10. Jianer Chen, Lihua Liu, and Weijia Jia. Improvement on vertex cover for low-degree graphs. Networks, 35(4):253-259, 2000. URL: https://doi.org/10.1002/1097-0037(200007)35:4%3C253::AID-NET3%3E3.0.CO;2-K.
  11. Anuj Dawar, Martin Grohe, and Stephan Kreutzer. Locally excluding a minor. In 22nd IEEE Symposium on Logic in Computer Science (LICS 2007), 10-12 July 2007, Wroclaw, Poland, Proceedings, pages 270-279. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/LICS.2007.31.
  12. Anuj Dawar and Stephan Kreutzer. Domination problems in nowhere-dense classes. In Ravi Kannan and K. Narayan Kumar, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2009, December 15-17, 2009, IIT Kanpur, India, volume 4 of LIPIcs, pages 157-168. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2009. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2009.2315.
  13. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi 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.
  14. Rodney G. Downey and Michael R. Fellows. Fixed parameter tractability and completeness III: some structural aspects of the W hierarchy. In Klaus Ambos-Spies, Steven Homer, and Uwe Schöning, editors, Complexity Theory: Current Research, Dagstuhl Workshop, February 2-8, 1992, pages 191-225. Cambridge University Press, 1992. Google Scholar
  15. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  16. Jan Dreier, Ioannis Eleftheriadis, Nikolas Mählmann, Rose McCarty, Michal Pilipczuk, and Szymon Torunczyk. First-order model checking on monadically stable graph classes. In 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, Chicago, IL, USA, October 27-30, 2024, pages 21-30. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00012.
  17. Jan Dreier, Daniel Mock, and Peter Rossmanith. Evaluating restricted first-order counting properties on nowhere dense classes and beyond. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 43:1-43:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.43.
  18. Jan Dreier and Peter Rossmanith. Approximate evaluation of first-order counting queries. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1720-1739. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.104.
  19. John A. Ellis, Hongbing Fan, and Michael R. Fellows. The dominating set problem is fixed parameter tractable for graphs of bounded genus. J. Algorithms, 52(2):152-168, 2004. URL: https://doi.org/10.1016/J.JALGOR.2004.02.001.
  20. Jörg Flum and Martin Grohe. Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput., 31(1):113-145, 2001. URL: https://doi.org/10.1137/S0097539799360768.
  21. Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana. FPT approximation and subexponential algorithms for covering few or many edges. Inf. Process. Lett., 185:106471, 2024. URL: https://doi.org/10.1016/J.IPL.2024.106471.
  22. Markus Frick and Martin Grohe. Deciding first-order properties of locally tree-decomposable structures. J. ACM, 48(6):1184-1206, 2001. URL: https://doi.org/10.1145/504794.504798.
  23. Jakub Gajarský, Michal Pilipczuk, Wojciech Przybyszewski, and Szymon Torunczyk. Twin-width and types. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 123:1-123:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ICALP.2022.123.
  24. Petr A. Golovach and Yngve Villanger. Parameterized complexity for domination problems on degenerate graphs. In Hajo Broersma, Thomas Erlebach, Tom Friedetzky, and Daniël Paulusma, editors, Graph-Theoretic Concepts in Computer Science, 34th International Workshop, WG 2008, Durham, UK, June 30 - July 2, 2008. Revised Papers, volume 5344 of Lecture Notes in Computer Science, pages 195-205, 2008. URL: https://doi.org/10.1007/978-3-540-92248-3_18.
  25. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 89-98. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591851.
  26. Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Parameterized complexity of vertex cover variants. Theory Comput. Syst., 41:501-520, October 2007. URL: https://doi.org/10.1007/s00224-007-1309-3.
  27. Mrinmoy Hota, Madhumangal Pal, and Tapan Kumar Pal. An efficient algorithm for finding a maximum weight k-independent set on trapezoid graphs. Comput. Optim. Appl., 18(1):49-62, 2001. URL: https://doi.org/10.1023/A:1008791627588.
  28. Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, and Saket Saurabh. Further exploiting c-closure for FPT algorithms and kernels for domination problems. SIAM J. Discret. Math., 37(4):2626-2669, 2023. URL: https://doi.org/10.1137/22M1491721.
  29. Joachim Kneis, Daniel Mölle, and Peter Rossmanith. Partial vs. complete domination: t-dominating set. In Jan van Leeuwen, Giuseppe F. Italiano, Wiebe van der Hoek, Christoph Meinel, Harald Sack, and Frantisek Plasil, editors, SOFSEM 2007: Theory and Practice of Computer Science, 33rd Conference on Current Trends in Theory and Practice of Computer Science, Harrachov, Czech Republic, January 20-26, 2007, Proceedings, volume 4362 of Lecture Notes in Computer Science, pages 367-376. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-69507-3_31.
  30. Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Exploiting c-closure in kernelization algorithms for graph problems. SIAM J. Discret. Math., 36(4):2798-2821, 2022. URL: https://doi.org/10.1137/21M1449476.
  31. Dietrich Kuske and Nicole Schweikardt. First-order logic with counting. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/LICS.2017.8005133.
  32. Vahan Mkrtchyan and Garik Petrosyan. On the fixed-parameter tractability of the partial vertex cover problem with a matching constraint in edge-weighted bipartite graphs. J. Graph Algorithms Appl., 26(1):91-110, 2022. URL: https://doi.org/10.7155/JGAA.00584.
  33. Vahan Mkrtchyan, Garik Petrosyan, K. Subramani, and Piotr Wojciechowski. On the partial vertex cover problem in bipartite graphs - a parameterized perspective. Theory Comput. Syst., 68(1):122-143, 2024. URL: https://doi.org/10.1007/S00224-023-10152-W.
  34. Daniel Mock and Peter Rossmanith. Solving a family of multivariate optimization and decision problems on classes of bounded expansion. In Hans L. Bodlaender, editor, 19th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2024, June 12-14, 2024, Helsinki, Finland, volume 294 of LIPIcs, pages 35:1-35:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.SWAT.2024.35.
  35. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  36. Rolf Niedermeier and Peter Rossmanith. Upper bounds for vertex cover further improved. In Christoph Meinel and Sophie Tison, editors, STACS 99, 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 4-6, 1999, Proceedings, volume 1563 of Lecture Notes in Computer Science, pages 561-570. Springer, 1999. URL: https://doi.org/10.1007/3-540-49116-3_53.
  37. Rolf Niedermeier and Peter Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms, 47(2):63-77, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00005-1.
  38. Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Solving dominating set in larger classes of graphs: FPT algorithms and polynomial kernels. In Amos Fiat and Peter Sanders, editors, Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, volume 5757 of Lecture Notes in Computer Science, pages 694-705. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_62.
  39. Detlef Seese. Linear time computable problems and first-order descriptions. Math. Struct. Comput. Sci., 6(6):505-526, 1996. URL: https://doi.org/10.1017/s0960129500070079.
  40. Ulrike Stege, Iris van Rooij, Alexander Hertel, and Philipp Hertel. An o(pn + 1.151^p)-algorithm for p-profit cover and its practical implications for vertex cover. In Prosenjit Bose and Pat Morin, editors, Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, volume 2518 of Lecture Notes in Computer Science, pages 249-261. Springer, 2002. URL: https://doi.org/10.1007/3-540-36136-7_23.
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