Document Open Access Logo

Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity

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

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2018.32.pdf
  • Filesize: 0.52 MB
  • 13 pages

Document Identifiers

Author Details

Daniel Lokshtanov
Pranabendu Misra
Fahad Panolan
Saket Saurabh
Meirav Zehavi

Cite AsGet BibTex

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ITCS.2018.32

Abstract

Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union representation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the transversal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r\in\mathbb{N}, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union~representation. Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element.
Keywords
  • travserval matroid
  • matroid representation
  • union representation
  • representative set

Metrics

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

References

  1. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Constrained multilinear detection and generalized graph motifs. Algorithmica, 74(2):947-967, 2016. URL: http://dx.doi.org/10.1007/s00453-015-9981-1.
  2. R A DeMillo and R J Lipton. A probabilistic remark on algebraic program testing. Inform. Process Lett., 7(4):193-195, 1978. Google Scholar
  3. Banu Dost, Tomer Shlomi, Nitin Gupta, Eytan Ruppin, Vineet Bafna, and Roded Sharan. Qnet: A tool for querying protein interaction networks. Journal of Computational Biology, 15(7):913-925, 2008. URL: http://dx.doi.org/10.1089/cmb.2007.0172.
  4. Stephen A. Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite perfect matching is in quasi-nc. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 754-763. ACM, 2016. URL: http://dx.doi.org/10.1145/2897518.2897564.
  5. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL: http://dx.doi.org/10.1145/2886094.
  6. François Le Gall. Powers of tensors and fast matrix multiplication. In Katsusuke Nabeshima, Kosaku Nagasaka, Franz Winkler, and Ágnes Szántó, editors, International Symposium on Symbolic and Algebraic Computation, ISSAC '14, Kobe, Japan, July 23-25, 2014, pages 296-303. ACM, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
  7. Shafi Goldwasser and Ofer Grossman. Bipartite perfect matching in pseudo-deterministic NC. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 87:1-87:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.87.
  8. Prachi Goyal, Neeldhara Misra, Fahad Panolan, and Meirav Zehavi. Deterministic algorithms for matching and packing problems based on representative sets. SIAM J. Discrete Math., 29(4):1815-1836, 2015. URL: http://dx.doi.org/10.1137/140981290.
  9. Sylvain Guillemot and Florian Sikora. Finding and counting vertex-colored subtrees. Algorithmica, 65(4):828-844, 2013. URL: http://dx.doi.org/10.1007/s00453-011-9600-8.
  10. Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in quasi-nc. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 821-830. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055440.
  11. Ioannis Koutis. Constrained multilinear detection for faster functional motif discovery. Inf. Process. Lett., 112(22):889-892, 2012. URL: http://dx.doi.org/10.1016/j.ipl.2012.08.008.
  12. Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. Deterministic truncation of linear matroids. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 922-934. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_75.
  13. Dániel Marx. A parameterized view on matroid optimization problems. Theor. Comput. Sci., 410(44):4471-4479, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.07.027.
  14. Pranabendu Misra, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Linear representation of transversal matroids and gammoids parameterized by rank. In Yixin Cao and Jianer Chen, editors, Computing and Combinatorics - 23rd International Conference, COCOON 2017, Hong Kong, China, August 3-5, 2017, Proceedings, volume 10392 of Lecture Notes in Computer Science, pages 420-432. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-62389-4_35.
  15. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL: http://dx.doi.org/10.1007/BF02579206.
  16. Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23-25 October 1995, pages 182-191. IEEE Computer Society, 1995. URL: http://dx.doi.org/10.1109/SFCS.1995.492475.
  17. J.G. Oxley. Matroid Theory. Oxford graduate texts in mathematics. Oxford University Press, 2006. Google Scholar
  18. Fahad Panolan and Meirav Zehavi. Parameterized algorithms for list k-cycle. In Akash Lal, S. Akshay, Saket Saurabh, and Sandeep Sen, editors, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016, December 13-15, 2016, Chennai, India, volume 65 of LIPIcs, pages 22:1-22:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2016.22.
  19. Ron Y. Pinter, Oleg Rokhlenko, Esti Yeger Lotem, and Michal Ziv-Ukelson. Alignment of metabolic pathways. Bioinformatics, 21(16):3401-3408, 2005. URL: http://dx.doi.org/10.1093/bioinformatics/bti554.
  20. Ron Y. Pinter, Hadas Shachnai, and Meirav Zehavi. Deterministic parameterized algorithms for the graph motif problem. Discrete Applied Mathematics, 213:162-178, 2016. URL: http://dx.doi.org/10.1016/j.dam.2016.04.026.
  21. Ron Y. Pinter and Meirav Zehavi. Algorithms for topology-free and alignment network queries. J. Discrete Algorithms, 27:29-53, 2014. URL: http://dx.doi.org/10.1016/j.jda.2014.03.002.
  22. J T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach., 27(4):701-717, 1980. Google Scholar
  23. Tomer Shlomi, Daniel Segal, Eytan Ruppin, and Roded Sharan. Qpath: a method for querying pathways in a protein-protein interaction network. BMC Bioinformatics, 7:199, 2006. URL: http://dx.doi.org/10.1186/1471-2105-7-199.
  24. Ola Svensson and Jakub Tarnawski. The matching problem in general graphs is in quasi-nc. CoRR, abs/1704.01929, 2017. URL: http://arxiv.org/abs/1704.01929.
  25. R Zippel. Probabilistic algorithms for sparse polynomials. In EUROSAM, pages 216-226, 1979. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact