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Improved Composition Theorems for Functions and Relations

Authors Sajin Koroth , Or Meir

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LIPIcs.APPROX-RANDOM.2018.48.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity classes
Keywords
  • circuit complexity
  • communication complexity
  • KRW conjecture
  • composition

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Abstract

One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P, i.e., to prove that P is not contained in NC^1. As an approach for this question, Karchmer, Raz and Wigderson [Mauricio Karchmer et al., 1995] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC^{1}.
Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [Edmonds et al., 2001], Håstad and Wigderson [Johan Håstad and Avi Wigderson, 1990] and Gavinsky, Meir, Weinstein and Wigderson [Dmitry Gavinsky et al., 2014] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.

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Sajin Koroth and Or Meir. Improved Composition Theorems for Functions and Relations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.48

Author Details

Sajin Koroth
  • Department of Computer Science, University of Haifa, Haifa 3498838, Israel
Or Meir
  • Department of Computer Science, University of Haifa, Haifa 3498838,Israel

Funding

  • Koroth, Sajin: Supported by the Israel Science Foundation (grant No. 1445/16)
  • Meir, Or: Partially supported by the Israel Science Foundation (grant No. 1445/16)

References

  1. J. Edmonds, R. Impagliazzo, S. Rudich, and J. Sgall. Communication complexity towards lower bounds on circuit depth. computational complexity, 10(3):210-246, Dec 2001. URL: http://dx.doi.org/10.1007/s00037-001-8195-x.
  2. Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward better formula lower bounds: an information complexity approach to the KRW composition conjecture. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 213-222, 2014. Google Scholar
  3. Johan Håstad and Avi Wigderson. Composition of the universal relation. In Advances In Computational Complexity Theory, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 119-134. DIMACS/AMS, 1990. Google Scholar
  4. Mauricio Karchmer, Eyal Kushilevitz, and Noam Nisan. Fractional covers and communication complexity. SIAM J. Discrete Math., 8(1):76-92, 1995. Google Scholar
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  7. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 234-243, 1997. Google Scholar
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  9. G. Tardos and U. Zwick. The communication complexity of the universal relation. In Proceedings of Computational Complexity. Twelfth Annual IEEE Conference, pages 247-259, Jun 1997. URL: http://dx.doi.org/10.1109/CCC.1997.612320.
  10. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209-213, 1979. Google Scholar
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