A Polynomial Kernel for Bipartite Permutation Vertex Deletion

Authors Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, Shaily Verma



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Author Details

Lawqueen Kanesh
  • National University of Singapore, Singapore
Jayakrishnan Madathil
  • Chennai Mathematical Institute, Chennai, India
Abhishek Sahu
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Shaily Verma
  • The Institute of Mathematical Sciences, HBNI, Chennai, India

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Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, and Shaily Verma. A Polynomial Kernel for Bipartite Permutation Vertex Deletion. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.IPEC.2021.23

Abstract

In a permutation graph, vertices represent the elements of a permutation, and edges represent pairs of elements that are reversed by the permutation. In the Permutation Vertex Deletion problem, given an undirected graph G and an integer k, the objective is to test whether there exists a vertex subset S ⊆ V(G) such that |S| ≤ k and G-S is a permutation graph. The parameterized complexity of Permutation Vertex Deletion is a well-known open problem. Bożyk et al. [IPEC 2020] initiated a study towards this problem by requiring that G-S be a bipartite permutation graph (a permutation graph that is bipartite). They called this the Bipartite Permutation Vertex Deletion (BPVD) problem. They showed that the problem admits a factor 9-approximation algorithm as well as a fixed parameter tractable (FPT) algorithm running in time 𝒪(9^k |V(G)|⁹). And they posed the question {whether BPVD admits a polynomial kernel.} 
We resolve this question in the affirmative by designing a polynomial kernel for BPVD. In particular, we obtain the following: Given an instance (G,k) of BPVD, in polynomial time we obtain an equivalent instance (G',k') of BPVD such that k' ≤ k, and |V(G')|+|E(G')| ≤ k^{𝒪(1)}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • kernelization
  • bipartite permutation graph
  • bicliques

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References

  1. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Feedback vertex set inspired kernel for chordal vertex deletion. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1383-1398, 2017. Google Scholar
  2. Akanksha Agrawal, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Interval vertex deletion admits a polynomial kernel. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1711-1730. SIAM, 2019. Google Scholar
  3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. Google Scholar
  4. Lukasz Bozyk, Jan Derbisz, Tomasz Krawczyk, Jana Novotná, and Karolina Okrasa. Vertex deletion into bipartite permutation graphs. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 180 of LIPIcs, pages 5:1-5:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  5. Andreas Brandstädt, Van Bang Le, and Jeremy P Spinrad. Graph classes: a survey. SIAM, 1999. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer-Verlag, 2015. Google Scholar
  7. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 68-81, 2012. Google Scholar
  8. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Journal of the ACM, 61(4):23:1-23:27, 2014. Google Scholar
  9. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  10. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  11. Andrew Drucker. New limits to classical and quantum instance compression. SIAM Journal on Computing, 44(5):1443-1479, 2015. Google Scholar
  12. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer-Verlag, Berlin, 2006. Google Scholar
  13. F. V. Fomin, S. Saurabh, and Y. Villanger. A Polynomial Kernel for Proper Interval Vertex Deletion. SIAM Journal on Discrete Mathematics, 27(4):1964-1976, 2013. Google Scholar
  14. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar f-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 470-479, 2012. Google Scholar
  15. Fedor V. Fomin and Saket Saurabh. Kernelization methods for fixed-parameter tractability. In Tractability, pages 260-282. Cambridge Univ. Press, Cambridge, 2014. Google Scholar
  16. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences, 77(1):91-106, 2011. Google Scholar
  17. Toshihiro Fujito. A unified approximation algorithm for node-deletion problems. Discrete Applied Mathematics, 86:213-231, 1998. Google Scholar
  18. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980. Google Scholar
  19. Jiong Guo and Rolf Niedermeier. Invitation to data reduction and problem kernelization. SIGACT News, 38(1):31-45, 2007. Google Scholar
  20. Danny Hermelin, Stefan Kratsch, Karolina Soltys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (turing) kernelization. Algorithmica, 71(3):702-730, 2015. Google Scholar
  21. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 104-113, 2012. Google Scholar
  22. Bart M. P. Jansen and Marcin Pilipczuk. Approximation and kernelization for chordal vertex deletion. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1399-1418, 2017. Google Scholar
  23. Yuping Ke, Yixin Cao, Xiating Ouyang, Wenjun Li, and Jianxin Wang. Unit interval vertex deletion: Fewer vertices are relevant. J. Comput. Syst. Sci., 95:109-121, 2018. Google Scholar
  24. Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. Google Scholar
  25. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. Google Scholar
  26. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, pages 129-161, 2012. Google Scholar
  27. Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM, 41:960-981, 1994. Google Scholar
  28. Dániel Marx. Chordal deletion is fixed-parameter tractable. Algorithmica, 57(4):747-768, 2010. Google Scholar
  29. Ross M. McConnell and Jeremy P. Spinrad. Modular decomposition and transitive orientation. Discret. Math., 201(1-3):189-241, 1999. Google Scholar
  30. Rolf Niedermeier. Invitation to fixed-parameter algorithms, volume 31. Oxford University Press, 2006. Google Scholar
  31. Jeremy P. Spinrad, Andreas Brandstädt, and Lorna Stewart. Bipartite permutation graphs. Discret. Appl. Math., 18(3):279-292, 1987. Google Scholar
  32. Ryuhei Uehara and Gabriel Valiente. Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett., 103(2):71-77, 2007. Google Scholar
  33. Mihalis Yannakakis. Node- and edge-deletion np-complete problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC), pages 253-264, 1978. Google Scholar
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