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FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

Authors Fedor V. Fomin , Petr A. Golovach , Tanmay Inamdar , Tomohiro Koana

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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tanmay Inamdar
  • University of Bergen, Norway
Tomohiro Koana
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Inamdar, Tanmay: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (LOPRE grant no. 819416).
  • Koana, Tomohiro: Supported by the DFG project DiPa (NI 369/21).

Cite AsGet BibTex

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana. FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 46:1-46:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.46

Abstract

We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ≤ α ≤ 1, the question is whether there is a set S ⊆ V of size k with a specified coverage function cov_α(S) at least p (or at most p for the minimization version). The coverage function cov_α(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1 - α. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut. A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α > 1/3 and minimization with α < 1/3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Partial Vertex Cover
  • Approximation Algorithms
  • Max Cut

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References

  1. Omid Amini, Fedor V. Fomin, and Saket Saurabh. Implicit Branching and Parameterized Partial Cover Problems. Journal of Computer and System Sciences, 77(6):1159-1171, 2011. Google Scholar
  2. Édouard Bonnet, Bruno Escoffier, Vangelis Th. Paschos, and Emeric Tourniaire. Multi-parameter Analysis for Local Graph Partitioning Problems: Using Greediness for Parameterization. Algorithmica, 71(3):566-580, 2015. Google Scholar
  3. Leizhen Cai. Parameterized Complexity of Cardinality Constrained Optimization Problems. The Computer Journal, 51(1):102-121, 2008. Google Scholar
  4. Derek G. Corneil and Yehoshua Perl. Clustering and domination in perfect graphs. Discret. Appl. Math., 9(1):27-39, 1984. URL: https://doi.org/10.1016/0166-218X(84)90088-X.
  5. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  6. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional parameters and local treewidth. SIAM J. Discret. Math., 18(3):501-511, 2004. URL: https://doi.org/10.1137/S0895480103433410.
  7. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Bidimensionality: new connections between FPT algorithms and PTASs. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 590-601. SIAM, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070514.
  8. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Ken-ichi Kawarabayashi. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23-25 October 2005, Pittsburgh, PA, USA, Proceedings, pages 637-646. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/SFCS.2005.14.
  9. Erik D. Demaine and MohammadTaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Comb., 28(1):19-36, 2008. URL: https://doi.org/10.1007/s00493-008-2140-4.
  10. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  11. Rodney G. Downey, Vladimir Estivill-Castro, Michael R. Fellows, Elena Prieto-Rodriguez, and Frances A. Rosamond. Cutting up is hard to do: the parameterized complexity of k-cut and related problems. In Computing: the Australasian Theory Symposiumm, CATS 2003, Adelaide, SA, Australia, February 4-7, 2003, volume 78 of Electronic Notes in Theoretical Computer Science, pages 209-222. Elsevier, 2003. URL: https://doi.org/10.1016/S1571-0661(04)81014-4.
  12. Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Contraction bidimensionality: The accurate picture. In Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, volume 5757 of Lecture Notes in Computer Science, pages 706-717. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_63.
  13. Fedor V. Fomin, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Subexponential parameterized algorithms for planar and apex-minor-free graphs via low treewidth pattern covering. SIAM J. Comput., 51(6):1866-1930, 2022. URL: https://doi.org/10.1137/19m1262504.
  14. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011. URL: https://doi.org/10.1016/j.ipl.2011.05.016.
  15. Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Parameterized Complexity of Vertex Cover Variants. Theory of Computing Systems, 41(3):501-520, 2007. Google Scholar
  16. Anupam Gupta, Euiwoong Lee, and Jason Li. Faster exact and approximate algorithms for k-cut. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 113-123. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00020.
  17. Anupam Gupta, Euiwoong Lee, and Jason Li. An FPT algorithm beating 2-approximation for k-cut. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2821-2837. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.179.
  18. J. Mark Keil and Timothy B. Brecht. The complexity of clustering in planar graphs. J. Combin. Math. Combin. Comput., 9:155-159, 1991. Google Scholar
  19. Tomohiro Koana, Christian Komusiewicz, André Nichterlein, and Frank Sommer. Covering Many (or Few) Edges with k Vertices in Sparse Graphs. In Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS '22), volume 219 of LIPIcs, pages 42:1-42:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  20. Pasin Manurangsi. A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation. In 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, volume 69 of OASIcs, pages 15:1-15:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.15.
  21. Pasin Manurangsi, Aviad Rubinstein, and Tselil Schramm. The strongish planted clique hypothesis and its consequences. In 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 10:1-10:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.10.
  22. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008. URL: https://doi.org/10.1093/comjnl/bxm048.
  23. Hadas Shachnai and Meirav Zehavi. Parameterized Algorithms for Graph Partitioning Problems. Theory of Computing Systems, 61(3):721-738, 2017. URL: https://doi.org/10.1007/s00224-016-9706-0.
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