Document Open Access Logo

Monotone Complexity of Spanning Tree Polynomial Re-Visited

Authors Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, Partha Mukhopadhyay



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2022.39.pdf
  • Filesize: 0.71 MB
  • 21 pages

Document Identifiers

Author Details

Arkadev Chattopadhyay
  • TIFR, Mumbai, India
Rajit Datta
  • Goldman-Sachs, Bangalore, India
Utsab Ghosal
  • Chennai Mathematical Institute, India
Partha Mukhopadhyay
  • Chennai Mathematical Institute, India

Acknowledgements

We thank the anonymous reviewers for their feedback.

Cite AsGet BibTex

Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay. Monotone Complexity of Spanning Tree Polynomial Re-Visited. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 39:1-39:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.39

Abstract

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having n variables and defined over constant-degree expander graphs, have monotone arithmetic complexity 2^{Ω(n)}. This yields the first strongly exponential lower bound on monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP [S. B. Gashkov and I. S. Sergeev, 2012; Ran Raz and Amir Yehudayoff, 2011; Srikanth Srinivasan, 2020; Bruno Pasqualotto Cavalar et al., 2020; Pavel Hrubeš and Amir Yehudayoff, 2021]. Recently, Hrubeš [Pavel Hrubeš, 2020] initiated a program to prove lower bounds against general arithmetic circuits by proving ε-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for ε ∈ (0,1). The first ε-sensitive lower bound was just proved for a family of polynomials inside VNP by Chattopadhyay, Datta and Mukhopadhyay [Arkadev Chattopadhyay et al., 2021]. We consider the spanning tree polynomial ST_n defined over the complete graph of n vertices and show that the polynomials F_{n-1,n} - ε⋅ ST_{n} and F_{n-1,n} + ε⋅ ST_{n}, defined over (n-1)n variables, have monotone circuit complexity 2^{Ω(n)} if ε ≥ 2^{- Ω(n)} and F_{n-1,n} := ∏_{i = 2}ⁿ (x_{i,1} + ⋯ + x_{i,n}) is the complete set-multilinear polynomial. This provides the first ε-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity. Our two results, thus, are incomparable generalizations of the well known result by Jerrum and Snir [Mark Jerrum and Marc Snir, 1982] which showed that the spanning tree polynomial, defined over complete graphs with n vertices (so the number of variables is (n-1)n), has monotone complexity 2^{Ω(n)}. In particular, the first result is an optimal lower bound and the second result can be thought of as a robust version of the earlier monotone lower bound for the spanning tree polynomial.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Spanning Tree Polynomial
  • Monotone Computation
  • Lower Bounds
  • Communication Complexity

Metrics

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

References

  1. Bruno Pasqualotto Cavalar, Mrinal Kumar, and Benjamin Rossman. Monotone circuit lower bounds from robust sunflowers. In Yoshiharu Kohayakawa and Flávio Keidi Miyazawa, editors, LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazil, January 5-8, 2021, Proceedings, volume 12118 of Lecture Notes in Computer Science, pages 311-322. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-61792-9_25.
  2. Arkadev Chattopadhyay, Rajit Datta, and Partha Mukhopadhyay. Lower bounds for monotone arithmetic circuits via communication complexity. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 786-799. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451069.
  3. Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput., 17(2):230-261, 1988. URL: https://doi.org/10.1137/0217015.
  4. S. B. Gashkov. On the complexity of monotone computations of polynomials. Vestn. Mosk. Univ., Ser. I, 1987(5), 1987. Google Scholar
  5. S. B. Gashkov and I. S. Sergeev. A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials. Sbornik. Mathematics, 203(10), 2012. Google Scholar
  6. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43:439-561, 2006. Google Scholar
  7. Pavel Hrubeš. On ε-sensitive monotone computations. Computational Complexity, 29(2):6, 2020. URL: https://doi.org/10.1007/s00037-020-00196-6.
  8. Pavel Hrubeš and Amir Yehudayoff. Shadows of newton polytopes. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 9:1-9:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.9.
  9. 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.
  10. O. M. Kasim-Zade. The complexity of monotone polynomials. Proceedings of the All-Union seminar on discrete mathematics and its applications (Russian) (Moscow, 1984), pages 136-138, 1986. Google Scholar
  11. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, USA, 2006. Google Scholar
  12. 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.
  13. Christofer Moore and Stephan Mertens. The Nature of Computation. Oxford University Press, 2011. Google Scholar
  14. Ran Raz and Amir Yehudayoff. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. J. Comput. Syst. Sci., 77(1):167-190, 2011. URL: https://doi.org/10.1016/j.jcss.2010.06.013.
  15. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics, 155(1):pp. 157-187, 2002. URL: http://www.jstor.org/stable/3062153.
  16. Srikanth Srinivasan. Strongly exponential separation between monotone VP and monotone VNP. Electron. Colloquium Comput. Complex., 26:32, 2019. URL: https://eccc.weizmann.ac.il/report/2019/032.
  17. Srikanth Srinivasan. Strongly exponential separation between monotone VP and monotone VNP. ACM Trans. Comput. Theory, 12(4):23:1-23:12, 2020. URL: https://doi.org/10.1145/3417758.
  18. Leslie G. Valiant. Negation can be exponentially powerful. Theor. Comput. Sci., 12:303-314, 1980. Preliminary version in STOC 1979. Google Scholar
  19. Moon J W. Counting labelled trees. Canadian Mathematical Congress, Montreal, 1970. Google Scholar
  20. Amir Yehudayoff. Separating monotone VP and VNP. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 425-429. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316311.
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