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Kernelization of Counting Problems

Authors Daniel Lokshtanov , Pranabendu Misra , Saket Saurabh , Meirav Zehavi

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Author Details

Daniel Lokshtanov
  • University of California Santa Barbara, CA, USA
Pranabendu Misra
  • Chennai Mathematical Institute, Chennai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

  • Lokshtanov, Daniel: Supported by NSF award CCF-2008838.
  • Saurabh, Saket: European Research Council (ERC) grant agreement no. 819416, and Swarnajayanti Fellowship no. DST/SJF/MSA01/2017-18.
  • Zehavi, Meirav: European Research Council (ERC) grant titled PARAPATH.

Cite AsGet BibTex

Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Kernelization of Counting Problems. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 77:1-77:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.77

Abstract

We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time either explicitly or by implicitly reducing the counting problems to enumeration problems. Thus, our framework is the only one in the spirit of classic kernelization (as well as lossy kernelization). Specifically, we define a compression of a counting problem P into a counting problem Q as a pair of polynomial-time procedures: reduce and lift. Given an instance of P, reduce outputs an instance of Q whose size is bounded by a function f of the parameter, and given the number of solutions to the instance of Q, lift outputs the number of solutions to the instance of P. When P = Q, compression is termed kernelization, and when f is polynomial, compression is termed polynomial compression. Our technical (and other conceptual) contributions can be classified into two categories: Upper Bounds. We prove two theorems: (i) The #Vertex Cover problem parameterized by solution size admits a polynomial kernel; (ii) Every problem in the class of #Planar F-Deletion problems parameterized by solution size admits a polynomial compression. Lower Bounds. We introduce two new concepts of cross-compositions: EXACT-cross-composition and SUM-cross-composition. We prove that if a #P-hard counting problem P EXACT-cross-composes into a parameterized counting problem Q, then Q does not admit a polynomial compression unless the polynomial hierarchy collapses. We conjecture that the same statement holds for SUM-cross-compositions. Then, we prove that: (i) #Min (s,t)-Cut parameterized by treewidth does not admit a polynomial compression unless the polynomial hierarchy collapses; (ii) #Min (s,t)-Cut parameterized by minimum cut size, #Odd Cycle Transversal parameterized by solution size, and #Vertex Cover parameterized by solution size minus maximum matching size, do not admit polynomial compressions unless our conjecture is false.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Kernelization
  • Counting Problems

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References

  1. Akanksha Agarwal, Saket Saurabh, and Prafullkumar Tale. On the parameterized complexity of contraction to generalization of trees. Theory of Computing Systems, 63(3):587-614, 2019. Google Scholar
  2. Pierre Bergé, Benjamin Mouscadet, Arpad Rimmel, and Joanna Tomasik. Fixed-parameter tractability of counting small minimum (s, t)-cuts. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 79-92. Springer, 2019. Google Scholar
  3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. Google Scholar
  4. Hans L Bodlaender, Bart MP Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. Google Scholar
  5. Marco Bressan and Marc Roth. Exact and approximate pattern counting in degenerate graphs: New algorithms, hardness results, and complexity dichotomies. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 276-285, 2021. Google Scholar
  6. Jonathan F Buss and Judy Goldsmith. Nondeterminism within p^. SIAM Journal on Computing, 22(3):560-572, 1993. Google Scholar
  7. Liming Cai, Jianer Chen, Rodney G Downey, and Michael R Fellows. Advice classes of parameterized tractability. Annals of pure and applied logic, 84(1):119-138, 1997. Google Scholar
  8. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms second edition. MIT Press and McGraw-Hill, 2001. Google Scholar
  9. Nadia Creignou, Arne Meier, Julian-Steffen Müller, Johannes Schmidt, and Heribert Vollmer. Paradigms for parameterized enumeration. Theory of Computing Systems, 60(4):737-758, 2017. Google Scholar
  10. Radu Curticapean. Counting problems in parameterized complexity. In 13th International Symposium on Parameterized and Exact Computation, IPEC 2018, August 20-24, 2018, Helsinki, Finland, pages 1:1-1:18, 2018. Google Scholar
  11. Radu Curticapean. A full complexity dichotomy for immanant families. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1770-1783, 2021. Google Scholar
  12. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223, 2017. Google Scholar
  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, New Orleans, LA, USA, January 7-10, 2018, pages 1080-1099, 2018. Google Scholar
  14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  15. Holger Dell, John Lapinskas, and Kitty Meeks. Approximately counting and sampling small witnesses using a colourful decision oracle. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2201-2211, 2020. Google Scholar
  16. Reinhard Diestel. Extremal graph theory. In Graph Theory, pages 173-207. Springer, 2017. Google Scholar
  17. Rodney G Downey and Michael R Fellows. Fundamentals of parameterized complexity, volume 4. Springer, 2013. Google Scholar
  18. Andrew Drucker. New limits to classical and quantum instance compression. SIAM Journal on Computing, 44(5):1443-1479, 2015. Google Scholar
  19. Eduard Eiben, Danny Hermelin, and MS Ramanujan. Lossy kernels for hitting subgraphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), pages 67:1-67:14. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  20. Eduard Eiben, Mithilesh Kumar, Amer E Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy kernels for connected dominating set on sparse graphs. SIAM Journal on Discrete Mathematics, 33(3):1743-1771, 2019. Google Scholar
  21. Michael R Fellows. The lost continent of polynomial time: Preprocessing and kernelization. In International Workshop on Parameterized and Exact Computation, pages 276-277. Springer, 2006. Google Scholar
  22. Michael R. Fellows, Ariel Kulik, Frances A. Rosamond, and Hadas Shachnai. Parameterized approximation via fidelity preserving transformations. J. Comput. Syst. Sci., 93:30-40, 2018. Google Scholar
  23. Jacob Focke and Marc Roth. Counting small induced subgraphs with hereditary properties. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1543-1551, 2022. Google Scholar
  24. Fedor V Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar f-deletion: Approximation, kernelization and optimal fpt algorithms. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 470-479. IEEE, 2012. Google Scholar
  25. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  26. Petr A. Golovach, Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Refined notions of parameterized enumeration kernels with applications to matching cut enumeration. J. Comput. Syst. Sci., 123:76-102, 2022. Google Scholar
  27. Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes for independent set of rectangles and geometric knapsack. In 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, pages 53:1-53:16, 2019. Google Scholar
  28. Akira Isihara. Statistical physics. Academic Press, 2013. Google Scholar
  29. Bart M. P. Jansen and Bart van der Steenhoven. Kernelization for counting problems on graphs: Preserving the number of minimum solutions. In Accepted to 18th IInternational Symposium on Parameterized and Exact Computation (IPEC 2023). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2023. Google Scholar
  30. Eun Jung Kim, Maria J. Serna, and Dimitrios M. Thilikos. Data-compression for parametrized counting problems on sparse graphs. In 29th International Symposium on Algorithms and Computation, ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan, pages 20:1-20:13, 2018. Google Scholar
  31. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. Journal of the ACM (JACM), 67(3):1-50, 2020. Google Scholar
  32. R Krithika, Diptapriyo Majumdar, and Venkatesh Raman. Revisiting connected vertex cover: Fpt algorithms and lossy kernels. Theory of Computing Systems, 62(8):1690-1714, 2018. Google Scholar
  33. Ramaswamy Krithika, Pranabendu Misra, Ashutosh Rai, and Prafullkumar Tale. Lossy kernels for graph contraction problems. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  34. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 224-237, 2017. Google Scholar
  35. Daniel Lokshtanov, Fahad Panolan, MS Ramanujan, and Saket Saurabh. Lossy kernelization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 224-237, 2017. Google Scholar
  36. Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Efficient computation of representative weight functions with applications to parameterized counting (extended version). In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 179-198, 2021. Google Scholar
  37. 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, pages 15:1-15:21, 2019. Google Scholar
  38. Ron Milo, Shai Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri Chklovskii, and Uri Alon. Network motifs: simple building blocks of complex networks. Science, 298(5594):824-827, 2002. Google Scholar
  39. J Scott Provan and Michael O Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing, 12(4):777-788, 1983. Google Scholar
  40. Dimitrios M Thilikos. Compactors for parameterized counting problems. Computer Science Review, 39:100344, 2021. Google Scholar
  41. Marc Thurley. Kernelizations for parameterized counting problems. In Theory and Applications of Models of Computation, 4th International Conference, TAMC 2007, Shanghai, China, May 22-25, 2007, Proceedings, pages 703-714, 2007. Google Scholar
  42. Leslie G Valiant. The complexity of computing the permanent. Theoretical computer science, 8(2):189-201, 1979. Google Scholar
  43. Leslie G Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410-421, 1979. Google Scholar
  44. René van Bevern, Till Fluschnik, and Oxana Yu Tsidulko. On approximate data reduction for the rural postman problem: Theory and experiments. Networks, 76(4):485-508, 2020. Google Scholar
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