Kernelization of Cycle Packing with Relaxed Disjointness Constraints

Authors Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, Saket Saurabh



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2016.26.pdf
  • Filesize: 0.69 MB
  • 14 pages

Document Identifiers

Author Details

Akanksha Agrawal
Daniel Lokshtanov
Diptapriyo Majumdar
Amer E. Mouawad
Saket Saurabh

Cite AsGet BibTex

Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, and Saket Saurabh. Kernelization of Cycle Packing with Relaxed Disjointness Constraints. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.26

Abstract

A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1.
Keywords
  • parameterized complexity
  • cycle packing
  • kernelization
  • relaxation

Metrics

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

References

  1. Hasan Abasi, Nader H. Bshouty, Ariel Gabizon, and Elad Haramaty. On r-simple k-path. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, pages 1-12, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_1.
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, December 1996. 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. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
  4. Hans L. Bodlaender and Arie M. C. A. Koster. Combinatorial optimization on graphs of bounded treewidth. Comput. J., 51(3):255-269, May 2008. Google Scholar
  5. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
  6. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H.
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  8. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 68-81, 2012. URL: http://portal.acm.org/citation.cfm?id=2095122&CFID=63838676&CFTOKEN=79617016.
  9. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: http://dx.doi.org/10.1145/2629620.
  10. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  11. Rod G. Downey and Michael R. Fellows. Parameterized complexity. Springer-Verlag, 1997. Google Scholar
  12. Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: http://dx.doi.org/10.1137/130927115.
  13. Paul Erdős and Lajos Pósa. On independent circuits contained in a graph. Canad. Journ. Math, 17(0):347-352, 1965. Google Scholar
  14. Henning Fernau, Alejandro López-Ortiz, and Jazmín Romero. Kernelization algorithms for packing problems allowing overlaps. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, Proceedings, pages 415-427, 2015. URL: http://dx.doi.org/10.1007/978-3-319-17142-5_35.
  15. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006. Google Scholar
  16. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
  17. Ariel Gabizon, Daniel Lokshtanov, and Michal Pilipczuk. Fast algorithms for parameterized problems with relaxed disjointness constraints. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 545-556, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_46.
  18. Danny Hermelin, Stefan Kratsch, Karolina Soltys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (turing) kernelization. Algorithmica, 71(3):702-730, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9910-8.
  19. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 104-113, 2012. URL: http://portal.acm.org/citation.cfm?id=2095125&CFID=63838676&CFTOKEN=79617016.
  20. Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
  21. 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. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_10.
  22. Daniel Lokshtanov, Fahad Panolan, M.S. Ramanujan, and Saket Saurabh. Lossy kernelization. arXiv:1604.04111, 2016. Google Scholar
  23. Rolf Niedermeier. Invitation to fixed-parameter algorithms. Oxford University Press, Oxford, 2006. Google Scholar
  24. Jan Ramon and Siegfried Nijssen. Polynomial-delay enumeration of monotonic graph classes. The Journal of Machine Learning Research, 10:907-929, 2009. Google Scholar
  25. Daniel Ratner and Manfred K. Warmuth. NxN puzzle and related relocation problem. J. Symb. Comput., 10(2):111-138, 1990. URL: http://dx.doi.org/10.1016/S0747-7171(08)80001-6.
  26. Jazmín Romero and Alejandro López-Ortiz. The 𝒢-packing with t-overlap problem. In Algorithms and Computation - 8th International Workshop, WALCOM 2014, Chennai, India, February 13-15, 2014, Proceedings, pages 114-124, 2014. URL: http://dx.doi.org/10.1007/978-3-319-04657-0_13.
  27. Jazmín Romero and Alejandro López-Ortiz. A parameterized algorithm for packing overlapping subgraphs. In Computer Science - Theory and Applications - 9th International Computer Science Symposium in Russia, CSR 2014, Moscow, Russia, June 7-11, 2014. Proceedings, pages 325-336, 2014. URL: http://dx.doi.org/10.1007/978-3-319-06686-8_25.
  28. Stéphan Thomassé. A 4k² kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, April 2010. URL: http://dx.doi.org/10.1145/1721837.1721848.
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