Abstract
In an unpublished Russian manuscript, Razborov proved that a matrix family with high rigidity over a finite field would yield a language outside the polynomial hierarchy in communication complexity. We present an alternative proof that strengthens the original result in several ways. In particular, we replace rigidity by the strictly weaker notion of toggle rigidity.
It turns out that Razborov’s astounding result is actually a corollary of a slight generalization of Toda’s First Theorem in communication complexity and that matrix rigidity over a finite field is a lower-bound method for bounded-error modular communication complexity. We also give evidence that Razborov’s strategy is a promising one by presenting a protocol with few alternations for the inner product function mod two and by discussing problems possibly outside the communication complexity version of the polynomial hierarchy.
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References
Noga Alon, Peter Frankl & Vojtech Rödl (1985). Geometrical Realization of Set Systems and Probabilistic Communication Complexity. In 26th Annual Symposium on Foundations of Computer Science, 21–23 October 1985, Portland, Oregon, USA, 277–280. IEEE Computer Society.
Dana Angluin (1980) On Counting Problems and the Polynomial- Time Hierarchy. Theor. Comput. Sci. 12: 161–173
Sanjeev Arora & Boaz Barak (2009). Computational Complexity. Cambridge University Press.
László Babai, Peter Frankl & Janos Simon (1986). Complexity classes in communication complexity theory (preliminary version). In 27th Annual Symposium on Foundations of Computer Science, FOCS 1986, 27–29 October 1986, Toronto, Ontario, Canada, 337–347. IEEE Computer Society.
José L. Balcázar, Josep Díaz & Joaquim Gabarró (1990). Structural Complexity II. Texts in Theoretical Computer Science, An EATCS Series. Springer-Verlag, 1st edition.
José L. Balcázar, Josep Díaz & Joaquim Gabarró (1995). Structural Complexity I. Texts in Theoretical Computer Science, An EATCS Series. Springer-Verlag, 2nd edition.
Richard Beigel, John Gill (1992) Counting Classes: Thresholds, Parity, Mods, and Fewness. Theor. Comput. Sci. 103(1): 3–23
Mahdi Cheraghchi (2005). On Matrix Rigidity and the Complexity of Linear Forms. Electronic Colloquium on Computational Complexity (ECCC) (070).
Fan Chung., Ronald Graham (2002) Sparse quasi-random graphs. Combinatorica 22: 217–244
Fan R.K. Chung, Ronald L. Graham, Richard M. Wilson (1989) Quasi-random graphs. Combinatorica 9(4): 345–362
Fan R.K. Chung., Prasad Tetali (1993) Communication Complexity and Quasi Randomness. SIAM J. Discrete Math. 6(1): 110–123
Bruno Codenotti (2000) Matrix rigidity. Linear Algebra and its Applications 304(1–3): 181–192
Bruno Codenotti., Pavel Pudlák., Giovanni Resta (2000) Some structural properties of low-rank matrices related to computational complexity. Theor. Comput. Sci. 235(1): 89–107
David Conlon (2008) A new upper bound for the bipartite Ramsey problem. Journal of Graph Theory 58(4): 351–356
Carsten Damm., Matthias Krause., Christoph Meinel., Stephan Waack (2004) On relations between counting communication complexity classes. J. Comput. Syst. Sci. 69(2): 259–280
Ding-Zhu Du & Ker-I Ko (2000). Theory of Computational Complexity. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., 1st edition.
Jürgen Forster (2002) A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci. 65(4): 612–625
Lance Fortnow (1997). Counting Complexity, 81–107. In Selman & Hemaspaandra (1997).
Joel Friedman (1993) A note on matrix rigidity. Combinatorica 13(2): 235–239
Lane A. Hemaspaandra & Mitsunori Ogihara (2002). The Complexity Theory Companion. Texts in Theoretical Computer Science, An EATCS Series. Springer-Verlag.
Bala Kalyanasundaram, Georg Schnitger (1992) The Probabilistic Communication Complexity of Set Intersection. SIAM J. Discrete Math. 5(4): 45–557
Hartmut Klauck (2003). Rectangle Size Bounds and Threshold Covers in Communication Complexity. In 18th Annual IEEE Conference on Computational Complexity,7–10 July 2003, Aarhus, Denmark, 118–134. IEEE Computer Society.
Johannes Köbler, Uwe Schöning & Jacobo Torán (1993). The Graph Isomorphism Problem – Its Structural Complexity. Birkhäuser Boston.
Michael Krivelevich & Benny Sudakov (2006). Pseudo-random graphs. In More sets, graphs and numbers, E. Györi, G. O. H. Katona & Laszlo Lovász, editors, volume 15 of Bolyai Soc. Math. Studies, 199–262. Springer-Verlag.
Eyal Kushilevitz & Noam Nisan (1997). Communication Complexity. Cambridge University Press.
Nathan Linial., Adi Shraibman (2009) Learning Complexity vs Communication Complexity. Combinatorics, Probability & Computing 18(1-2): 227–245
Nati Linial., Adi Shraibman (2009) Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34(3): 368–394
Satyanarayana V. Lokam (1996). Spectral Methods for Matrix Rigidity with Applications to Size-Depth Trade-offs and Communication Complexity. Electronically available at CiteSeerX, http://130.203.133.150/viewdoc/summary?doi=10.1.1.52.4411.
Satyanarayana V. Lokam (2000) On the rigidity of Vandermonde matrices. Theor. Comput. Sci. 237(1–2): 477–483
Satyanarayana V. Lokam (2001) Spectral Methods for Matrix Rigidity with Applications to Size-Depth Trade-offs and Communication Complexity. J. Comput. Syst. Sci. 63(3): 449–473
Satyanarayana V. Lokam (2006). Quadratic Lower Bounds on Matrix Rigidity. In Theory and Applications of Models of Computation, Third International Conference, TAMC 2006, Beijing, China, May 15–20, 2006, Proceedings, Jin-Yi Cai, S. Barry Cooper & Angsheng Li, editors, volume 3959 of Lecture Notes in Computer Science, 295–307. Springer-Verlag.
Satyanarayana V. Lokam (2009) Complexity Lower Bounds using Linear Algebra. Foundations and Trends in Theoretical Computer Science 4(1–2): 1–155
Gatis Midrijanis (2005). Three lines proof of the lower bound for the matrix rigidity. CoRR abs/cs/0506081.
Alon Orlitsky., Abbas El Gamal (1990) Average and randomized communication complexity. IEEE Transactions on Information Theory 36(1): 3–16
Christos H. Papadimitriou & Stathis Zachos (1983). Two remarks on the power of counting. In Theoretical Computer Science, 6th GI-Conference, Dortmund, Germany, January 5–7, 1983, Proceedings, Armin B. Cremers & Hans-Peter Kriegel, editors, volume 145 of Lecture Notes in Computer Science, 269–276. Springer-Verlag.
Pavel Pudlák (1994) Communication in Bounded Depth Circuits. Combinatorica 14(2): 203–216
Pavel Pudlák & Vojtech Rödl (1994) Some combinatorialalgebraic problems from complexity theory. Discrete Mathematics 136(1–3): 253–279
Alexander Razborov (1989). On Rigid Matrices (in Russian). Technical report, Steklov Mathematical Institute. Electronically available at http://people.cs.uchicago.edu/~razborov/rigid.pdf.
Alexander A. Razborov (1992) On the Distributional Complexity of Disjointness. Theor. Comput. Sci. 106(2): 385–390
Uwe Schöning (1986). Complexity and Structure, volume 211 of Lecture Notes in Computer Science. Springer-Verlag.
Uwe Schöning (1988). The power of counting. In Selman (1988), 204–223.
Uwe Schöning (1989) Probabilistic Complexity Classes and Lowness. J. Comput. Syst. Sci. 39(1): 84–100
Uwe Schöning (1991). Recent Highlights in Structural Complexity Theory (invited talk). In SOFSEM’91, Nizké Tratry (CSFR), Conference Proceedings, 205–216. Springer-Verlag.
Alan L. Selman (editor) (1988). Complexity Theory Retrospective. Foundations of Computing. Springer-Verlag.
Alan L. Selman & Lane A. Hemaspaandra (editors) (1997). Complexity Theory Retrospective II. Foundations of Computing. Springer- Verlag.
Alexander A. Sherstov (2008a). Communication Complexity under Product and Nonproduct Distributions. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC 2008, 23–26 June 2008, College Park, Maryland, USA, 64–70. IEEE Computer Society.
Alexander A. Sherstov (2008) Halfspace Matrices. Computational Complexity 17(2): 149–178
Mohammad Amin Shokrollahi, Daniel A. Spielman., Volker Stemann (1997) A Remark on Matrix Rigidity. Inf. Process. Lett. 64(6): 283–285
Seinosuke Toda (1990). Computational Complexity of Counting Complexity Classes. Ph.D. thesis, Dept. of Comput. Sci. & Inf. Mat., Univ. of Electro-Commun., Tokyo.
Seinosuke Toda (1991) PP is as Hard as the Polynomial-Time Hierarchy. SIAM J. Comput. 20(5): 865–877
Leslie G. Valiant (1977). Graph-Theoretic Arguments in Low-Level Complexity. In Mathematical Foundations of Computer Science 1977, 6th Symposium, Tatranska Lomnica, Czechoslovakia, September 5–9, 1977, Proceedings, Jozef Gruska, editor, volume 53 of Lecture Notes in Computer Science, 162–176. Springer-Verlag.
Leslie G. Valiant., Vijay V. Vazirani (1986) NP is as Easy as Detecting Unique Solutions. Theor. Comput. Sci. 47(3): 85–93
Ronald de Wolf (2006). Lower Bounds on Matrix Rigidity Via a Quantum Argument. In Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10– 14, 2006, Proceedings, Part I, Michele Bugliesi, Bart Preneel, Vladimiro Sassone & Ingo Wegener, editors, volume 4051 of Lecture Notes in Computer Science, 62–71. Springer-Verlag.
Henning Wunderlich (2009). On Toda’s Theorem in Structural Communication Complexity. In SOFSEM 2009: Theory and Practice of Computer Science, 35th Conference on Current Trends in Theory and Practice of Computer Science, Spindleruv Mlýn, Czech Republic, January 24–30, 2009. Proceedings, Mogens Nielsen, Antonín Kucera, Peter Bro Miltersen, Catuscia Palamidessi, Petr Tuma & Frank D. Valencia, editors, volume 5404 of Lecture Notes in Computer Science, 609–620. Springer-Verlag.
Andrew Chi-Chih Yao (1979). Some Complexity Questions Related to Distributive Computing (Preliminary Report). In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing, 30 April–2 May, 1979, Atlanta, Georgia, USA, 209–213. ACM.
Andrew Chi-Chih Yao (1983). Lower Bounds by Probabilistic Arguments (Extended Abstract). In 24th Annual Symposium on Foundations of Computer Science, FOCS 1983, 7–9 November 1983, Tucson, Arizona, USA, 420–428. IEEE Computer Society.
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Wunderlich, H. On a Theorem of Razborov. comput. complex. 21, 431–477 (2012). https://doi.org/10.1007/s00037-011-0021-5
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DOI: https://doi.org/10.1007/s00037-011-0021-5
Keywords
- Structural complexity
- communication complexity
- Toda’s theorems
- theorem of Razborov
- matrix rigidity
- approximate rank
- quasi-random graph