Document Open Access Logo

Deterministic Constrained Multilinear Detection

Authors Cornelius Brand, Viktoriia Korchemna, Michael Skotnica

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2023.25.pdf
  • Filesize: 0.66 MB
  • 14 pages

Document Identifiers

Author Details

Cornelius Brand
  • Algorithms and Complexity Group, TU Wien, Austria
Viktoriia Korchemna
  • Algorithms and Complexity Group, TU Wien, Austria
Michael Skotnica
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

  • Brand, Cornelius: Austrian Science Fund (FWF, project Y1329).
  • Korchemna, Viktoriia: Austrian Science Fund (FWF, project Y1329).
  • Skotnica, Michael: "Grant Schemes at CU" (reg. no. CZ.02.2.69/0.0/0.0/19_073/0016935), GAČR grant 22-19073S.

Cite AsGet BibTex

Cornelius Brand, Viktoriia Korchemna, and Michael Skotnica. Deterministic Constrained Multilinear Detection. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.25

Abstract

We extend the algebraic techniques of Brand and Pratt (ICALP'21) for deterministic detection of k-multilinear monomials in a given polynomial with non-negative coefficients to the more general situation of detecting colored k-multilinear monomials that satisfy additional constraints on the multiplicities of the colors appearing in them. Our techniques can be viewed as a characteristic-zero generalization of the algebraic tools developed by Guillemot and Sikora (MFCS'10) and Björklund, Kaski and Kowalik (STACS'13) As applications, we recover the state-of-the-art deterministic algorithms for the Graph Motif problem due to Pinter, Schachnai and Zehavi (MFCS'14), and give new deterministic algorithms for generalizations of certain questions on colored directed spanning trees or bipartite planar matchings running in deterministic time O^∗(4^k), studied originally by Gutin, Reidl, Wahlström and Zehavi (J. Comp. Sys. Sci. 95, '18). Finally, we give improved randomized algorithms for intersecting three and four matroids of rank k in characteristic zero, improving the record bounds of Brand and Pratt (ICALP'21) from O^∗(64^k) and O^∗(256^k), respectively, to O^∗(4^k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Fixed-parameter algorithms
  • Algebraic algorithms
  • Motif discovery
  • Matroid intersection

Metrics

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

References

  1. Alexander I Barvinok. New algorithms for linear k-matroid intersection and matroid k-parity problems. Mathematical Programming, 69:449-470, 1995. Google Scholar
  2. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 67-74. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250801.
  3. Andreas Björklund, Petteri Kaski, and Ioannis Koutis. Directed hamiltonicity and out-branchings via generalized laplacians. 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 91:1-91:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.91.
  4. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Probably optimal graph motifs. In Natacha Portier and Thomas Wilke, editors, 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 20-31. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2013. URL: https://doi.org/10.4230/LIPIcs.STACS.2013.20.
  5. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Constrained multilinear detection and generalized graph motifs. Algorithmica, 74(2):947-967, 2016. URL: https://doi.org/10.1007/s00453-015-9981-1.
  6. Cornelius Brand. Patching colors with tensors. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 25:1-25:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.25.
  7. Cornelius Brand. A note on algebraic techniques for subgraph detection. Inf. Process. Lett., 176:106242, 2022. URL: https://doi.org/10.1016/j.ipl.2021.106242.
  8. Cornelius Brand and Kevin Pratt. Parameterized applications of symbolic differentiation of (totally) multilinear polynomials. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 38:1-38:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.38.
  9. 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.
  10. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL: https://doi.org/10.1016/0020-0190(78)90067-4.
  11. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  12. Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, and Nitin Saurabh. Homomorphism polynomials complete for VP. Chic. J. Theor. Comput. Sci., 2016, 2016. URL: http://cjtcs.cs.uchicago.edu/articles/2016/3/contents.html.
  13. Jack Edmonds. Matroid intersection. In Annals of discrete Mathematics, volume 4, pages 39-49. Elsevier, 1979. Google Scholar
  14. Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. Determinantal sieving, 2023. URL: https://arxiv.org/abs/2304.02091.
  15. 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: https://doi.org/10.1145/2886094.
  16. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Representative families of product families. ACM Trans. Algorithms, 13(3):36:1-36:29, 2017. URL: https://doi.org/10.1145/3039243.
  17. Sylvain Guillemot and Florian Sikora. Finding and counting vertex-colored subtrees. In Petr Hlinený and Antonín Kucera, editors, Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings, volume 6281 of Lecture Notes in Computer Science, pages 405-416. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15155-2_36.
  18. Gregory Z. Gutin, Felix Reidl, Magnus Wahlström, and Meirav Zehavi. Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials. J. Comput. Syst. Sci., 95:69-85, 2018. URL: https://doi.org/10.1016/j.jcss.2018.01.004.
  19. Mark Jerrum and Marc Snir. Some exact complexity results for straight-line computations over semirings. J. ACM, 29(3):874-897, 1982. URL: https://doi.org/10.1145/322326.322341.
  20. Ioannis Koutis. Faster algebraic algorithms for path and packing problems. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, volume 5125 of Lecture Notes in Computer Science, pages 575-586. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-70575-8_47.
  21. Ioannis Koutis. Constrained multilinear detection for faster functional motif discovery. Inf. Process. Lett., 112(22):889-892, 2012. URL: https://doi.org/10.1016/j.ipl.2012.08.008.
  22. Ioannis Koutis and Ryan Williams. Limits and applications of group algebras for parameterized problems. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris E. Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, volume 5555 of Lecture Notes in Computer Science, pages 653-664. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_54.
  23. Meena Mahajan and V. Vinay. A combinatorial algorithm for the determinant. In Michael E. Saks, editor, Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 5-7 January 1997, New Orleans, Louisiana, USA, pages 730-738. ACM/SIAM, 1997. URL: http://dl.acm.org/citation.cfm?id=314161.314429.
  24. Dániel Marx. A parameterized view on matroid optimization problems. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, volume 4051 of Lecture Notes in Computer Science, pages 655-666. Springer, 2006. URL: https://doi.org/10.1007/11786986_57.
  25. Burkhard Monien. How to find long paths efficiently. In North-Holland Mathematics Studies, volume 109, pages 239-254. Elsevier, 1985. Google Scholar
  26. Ron Y. Pinter, Hadas Shachnai, and Meirav Zehavi. Deterministic parameterized algorithms for the graph motif problem. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, volume 8635 of Lecture Notes in Computer Science, pages 589-600. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44465-8_50.
  27. Alexander Postnikov. Total positivity, grassmannians, and networks, 2006. URL: https://arxiv.org/abs/math/0609764.
  28. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  29. Ryan Williams. Finding paths of length k in o^*(2^k) time. Inf. Process. Lett., 109(6):315-318, 2009. URL: https://doi.org/10.1016/j.ipl.2008.11.004.
  30. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Edward W. Ng, editor, Symbolic and Algebraic Computation, EUROSAM '79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings, volume 72 of Lecture Notes in Computer Science, pages 216-226. Springer, 1979. URL: https://doi.org/10.1007/3-540-09519-5_73.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact