33rd Computational Complexity Conference (CCC 2018), CCC 2018, June 22-24, 2018, San Diego, CA, USA
CCC 2018
June 22-24, 2018
San Diego, CA, USA
Computational Complexity Conference
CCC
http://computationalcomplexity.org
https://dblp.org/db/conf/coco
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Rocco A.
Servedio
Rocco A. Servedio
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
102
2018
978-3-95977-069-9
https://www.dagstuhl.de/dagpub/978-3-95977-069-9
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xi
Front Matter
Rocco A.
Servedio
Rocco A. Servedio
10.4230/LIPIcs.CCC.2018.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Pseudorandom Generators from Polarizing Random Walks
We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1,1]^n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1,1]^n that converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.
AC0
branching program
polarization
pseudorandom generators
random walks
Sensitivity
Theory of computation~Pseudorandomness and derandomization
1:1-1:21
Regular Paper
Eshan
Chattopadhyay
Eshan Chattopadhyay
Cornell Univeristy and IAS, Princeton, USA
Supported by NSF grant CCF-1412958 and the Simons foundation.
Pooya
Hatami
Pooya Hatami
University of Texas at Austin, USA
Supported by a Simons Investigator Award (#409864, David Zuckerman).
Kaave
Hosseini
Kaave Hosseini
University of California, San Diego, USA
Supported by NSF grant CCF-1614023.
Shachar
Lovett
Shachar Lovett
University of California, San Diego, USA
Supported by NSF grant CCF-1614023.
10.4230/LIPIcs.CCC.2018.1
Miklós Ajtai and Avi Wigderson. Deterministic simulation of probabilistic constant depth circuits. In Foundations of Computer Science, 1985., 26th Annual Symposium on, pages 11-19. IEEE, 1985.
Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple constructions of almost k-wise independent random variables. Random Structures &Algorithms, 3(3):289-304, 1992.
Nikhil Bansal. Constructive algorithms for discrepancy minimization. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 3-10. IEEE, 2010.
Mark Braverman. Polylogarithmic independence fools ac 0 circuits. Journal of the ACM (JACM), 57(5):28, 2010.
Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. Improved pseudorandomness for unordered branching programs through local monotonicity. In Electronic Colloquium on Computational Complexity (ECCC), pages TR17-171, 2017.
Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, and Salil Vadhan. Better pseudorandom generators from milder pseudorandom restrictions. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 120-129. IEEE, 2012.
Parikshit Gopalan, Rocco A Servedio, and Avi Wigderson. Degree and sensitivity: tails of two distributions. In Proceedings of the 31st Conference on Computational Complexity, page 13. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016.
Johan Hastad. Almost optimal lower bounds for small depth circuits. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 6-20. ACM, 1986.
Pooya Hatami and Avishay Tal. Pseudorandom generators for low-sensitivity functions. In Electronic Colloquium on Computational Complexity (ECCC), volume 24, page 25, 2017.
Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006.
Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, fourier transform, and learnability. Journal of the ACM (JACM), 40(3):607-620, 1993.
Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 61-67. IEEE, 2012.
Shachar Lovett, Avishay Tal, and Jiapeng Zhang. Robust sensitivity. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 161, 2016.
Yishay Mansour. An n^O(log log n) learning algorithm for dnf under the uniform distribution. Journal of Computer and System Sciences, 50(3):543-550, 1995.
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM journal on computing, 22(4):838-856, 1993.
Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991.
Noam Nisan and Avi Wigderson. Hardness vs. randomness. In Foundations of Computer Science, 1988., 29th Annual Symposium on, pages 2-11. IEEE, 1988.
Omer Reingold, Thomas Steinke, and Salil Vadhan. Pseudorandomness for regular branching programs via fourier analysis. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 655-670. Springer, 2013.
Hans-Ulrich Simon. A tight ω (loglog n)-bound on the time for parallel ram’s to compute nondegenerated boolean functions. In International Conference on Fundamentals of Computation Theory, pages 439-444. Springer, 1983.
Avishay Tal. Tight bounds on the fourier spectrum of ac0. In LIPIcs-Leibniz International Proceedings in Informatics, volume 79. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
Luca Trevisan and Tongke Xue. A derandomized switching lemma and an improved derandomization of ac0. In Computational Complexity (CCC), 2013 IEEE Conference on, pages 242-247. IEEE, 2013.
Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. Computational Complexity, 18(2):209-217, 2009.
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n
We construct and analyze a pseudorandom generator for degree 2 boolean polynomial threshold functions. Random constructions achieve the optimal seed length of O(log n + log 1/epsilon), however the best known explicit construction of [Ilias Diakonikolas, 2010] uses a seed length of O(log n * epsilon^{-8}). In this work we give an explicit construction that uses a seed length of O(log n + (1/epsilon)^{o(1)}). Note that this improves the seed length substantially and that the dependence on the error epsilon is additive and only grows subpolynomially as opposed to the previously known multiplicative polynomial dependence.
Our generator uses dimensionality reduction on a Nisan-Wigderson based pseudorandom generator given by Lu, Kabanets [Kabanets and Lu, 2018].
Pseudorandomness
Polynomial Threshold Functions
Theory of computation~Pseudorandomness and derandomization
2:1-2:24
Regular Paper
Daniel
Kane
Daniel Kane
UC San Diego
Supported by NSF Award CCF-1553288(CAREER) and a Sloan Research Fellowship.
Sankeerth
Rao
Sankeerth Rao
UC San Diego
10.4230/LIPIcs.CCC.2018.2
J. Wright A. Carbery. Distributional and l^q norm inequalities for polynomials over convex bodies in ℝⁿ. Mathematical Research Letters, 2001.
Richard Beigel. The polynomial method in circuit complexity. In Proceedings of the Eigth Annual Structure in Complexity Theory Conference, San Diego, CA, USA, May 18-21, 1993, pages 82-95. IEEE Computer Society, 1993. URL: http://dx.doi.org/10.1109/SCT.1993.336538.
http://dx.doi.org/10.1109/SCT.1993.336538
Anindya De, Ilias Diakonikolas, and Rocco A. Servedio. Deterministic approximate counting for degree-dollar2dollar polynomial threshold functions. CoRR, abs/1311.7105, 2013. URL: http://arxiv.org/abs/1311.7105.
http://arxiv.org/abs/1311.7105
Anindya De and Rocco A. Servedio. Efficient deterministic approximate counting for low-degree polynomial threshold functions. CoRR, abs/1311.7178, 2013. URL: http://arxiv.org/abs/1311.7178.
http://arxiv.org/abs/1311.7178
I. Diakonikolas, P. Gopalan, R. Jaiswal, R.A. Servedio, and E. Viola. Bounded independence fools halfspaces. SIAM Journal on Computing, 2010.
Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, and Andrew Wan. A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions. CoRR, abs/0909.4727, 2009. URL: http://arxiv.org/abs/0909.4727.
http://arxiv.org/abs/0909.4727
Parikshit Gopalan, Daniel M. Kane, and Raghu Meka. Pseudorandomness via the discrete fourier transform. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 903-922. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.60.
http://dx.doi.org/10.1109/FOCS.2015.60
Jelani Nelson Ilias Diakonikolas, Daniel M. Kane. Bounded independence fools degree-2 threshold functions. Foundations of Computer Science (FOCS), 2010.
Daniel M. Kane. k-independent gaussians fool polynomial threshold functions. Conference on Computational Complexity (CCC), 2011.
Daniel M. Kane. A small prg for polynomial threshold functions of gaussians. Symposium on the Foundations Of Computer Science (FOCS), 2011.
Daniel M. Kane. A structure theorem for poorly anticoncentrated gaussian chaoses and applications to the study of polynomial threshold functions. CORR, 2012. URL: http://arxiv.org/abs/1204.0543.
http://arxiv.org/abs/1204.0543
Daniel M. Kane. A pseudorandom generator for polynomial threshold functions of gaussians with subpolynomial seed length. Conference on Computational Complexity (CCC), 2014.
Daniel M. Kane. A polylogarithmic PRG for degree 2 threshold functions in the gaussian setting. In David Zuckerman, editor, 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, volume 33 of LIPIcs, pages 567-581. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.567.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.567
Adam R. Klivans and Rocco A. Servedio. Learning DNF in time 2^õ(n^1/3). In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 258-265. ACM, 2001. URL: http://dx.doi.org/10.1145/380752.380809.
http://dx.doi.org/10.1145/380752.380809
Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. CoRR, abs/0910.4122, 2009. URL: http://arxiv.org/abs/0910.4122.
http://arxiv.org/abs/0910.4122
E. Mossel, R. O'Donnell, and K. Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. ArXiv Mathematics e-prints, 2005. URL: http://arxiv.org/abs/math/0503503.
http://arxiv.org/abs/math/0503503
Alexander A. Sherstov. Separating ac^0 from depth-2 majority circuits. 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 294-301. ACM, 2007. URL: http://dx.doi.org/10.1145/1250790.1250834.
http://dx.doi.org/10.1145/1250790.1250834
Valentine Kabanets and Zhenjian Lu. Nisan-wigderson pseudorandom generators for circuits with polynomial threshold gates. ECCC, https://eccc.weizmann.ac.il/report 012/, 2018. URL: https://eccc.weizmann.ac.il/report/2018/012/.
https://eccc.weizmann.ac.il/report/2018/012/
Daniel Kane and Sankeerth Rao
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A New Approach for Constructing Low-Error, Two-Source Extractors
Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework (STOC 2016), and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient there and in follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck.
The parameters we require from t-non-malleable extractors for our reduction to work hold in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors.
Two-Source Extractors
Non-Malleable Extractors
Pseudorandomness
Explicit Constructions
Theory of computation~Pseudorandomness and derandomization
3:1-3:19
Regular Paper
Avraham
Ben-Aroya
Avraham Ben-Aroya
The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel
Supported by the Israel Science Foundation grant no. 994/14.
Eshan
Chattopadhyay
Eshan Chattopadhyay
Department of Computer Science, Cornell University and School of Mathematics, IAS, Ithaca, NY 14850, USA
Princeton, NJ 08540, USA
Supported by NSF grants CCF-1526952, CCF-1412958 and the Simons Foundation. Part of this work was done when the author was a graduate student in UT Austin and while visiting the Simons Institute for the Theory of Computing at UC Berkeley.
Dean
Doron
Dean Doron
The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel
Supported by the Israel Science Foundation grant no. 994/14. This work was done in part while visiting the Simons Institute for the Theory of Computing at UC Berkeley.
Xin
Li
Xin Li
Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218, USA
Supported by NSF Grant CCF-1617713.
Amnon
Ta-Shma
Amnon Ta-Shma
The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel
Supported by the Israel Science Foundation grant no. 994/14. This work was done in part while visiting the Simons Institute for the Theory of Computing at UC Berkeley.
10.4230/LIPIcs.CCC.2018.3
H.L. Abbott. Lower bounds for some Ramsey numbers. Discrete Mathematics, 2(4):289-293, 1972.
Noga Alon. The Shannon capacity of a union. Combinatorica, 18(3):301-310, 1998.
Boaz Barak. A simple explicit construction of an n^õ(log n)-Ramsey graph. arXiv preprint math/0601651, 2006.
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, and Avi Wigderson. Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. Journal of the ACM (JACM), 57(4):20, 2010.
Boaz Barak, Anup Rao, Ronen Shaltiel, and Avi Wigderson. 2-source dispersers for n^o(1) entropy, and Ramsey graphs beating the frankl-wilson construction. Annals of Mathematics, 176(3):1483-1544, 2012.
Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. An efficient reduction from two-source to non-malleable extractors: achieving near-logarithmic min-entropy. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC), pages 1185-1194. ACM, 2017.
Jean Bourgain. More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory, 1(01):1-32, 2005.
Eshan Chattopadhyay, Vipul Goyal, and Xin Li. Non-malleable extractors and codes, with their many tampered extensions. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 285-298. ACM, 2016.
Eshan Chattopadhyay and Xin Li. Explicit non-malleable extractors, multi-source extractors, and almost optimal privacy amplification protocols. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 158-167. IEEE, 2016.
Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 670-683. ACM, 2016.
Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988.
Fan R.K. Chung. A note on constructive methods for Ramsey numbers. Journal of Graph Theory, 5(1):109-113, 1981.
Gil Cohen. Local correlation breakers and applications to three-source extractors and mergers. SIAM Journal on Computing, 45(4):1297-1338, 2016.
Gil Cohen. Making the most of advice: New correlation breakers and their applications. In Proceedings of 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 188-196. IEEE, 2016.
Gil Cohen. Non-malleable extractors - new tools and improved constructions. In LIPIcs-Leibniz International Proceedings in Informatics, volume 50. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016.
Gil Cohen. Non-malleable extractors with logarithmic seeds. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 30, 2016.
Gil Cohen. Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 278-284. ACM, 2016.
Gil Cohen. Two-source extractors for quasi-logarithmic min-entropy and improved privacy amplification protocols. In ECCC, 2016.
Gil Cohen, Ran Raz, and Gil Segev. Nonmalleable extractors with short seeds and applications to privacy amplification. SIAM Journal on Computing, 43(2):450-476, 2014.
Gil Cohen and Igor Shinkar. Personal communication, 2017.
Yevgeniy Dodis and Daniel Wichs. Non-malleable extractors and symmetric key cryptography from weak secrets. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pages 601-610. ACM, 2009.
Peter Frankl. A constructive lower bound for ramsey numbers. Ars Combinatoria, 3(297-302):28, 1977.
Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981.
Vince Grolmusz. Low rank co-diagonal matrices and Ramsey graphs. Journal of combinatorics, 7(1):R15-R15, 2001.
Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on boolean functions. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science (FOCS), pages 68-80. IEEE, 1988.
Xin Li. Non-malleable extractors, two-source extractors and privacy amplification. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 688-697. IEEE, 2012.
Xin Li. Improved two-source extractors, and affine extractors for polylogarithmic entropy. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 168-177. IEEE, 2016.
Xin Li. Improved non-malleable extractors, non-malleable codes and independent source extractors. 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 1144-1156. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055486.
http://dx.doi.org/10.1145/3055399.3055486
Raghu Meka. Explicit resilient functions matching ajtai-linial. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1132-1148. SIAM, 2017.
Zs Nagy. A constructive estimation of the ramsey numbers. Mat. Lapok, 23:301-302, 1975.
Moni Naor. Constructing Ramsey graphs from small probability spaces. IBM Research Report RJ, 8810, 1992.
Victor Neumann-Lara. The dichromatic number of a digraph. Journal of Combinatorial Theory, Series B, 33(3):265-270, 1982.
Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24, 2000.
Ran Raz. Extractors with weak random seeds. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 11-20. ACM, 2005.
David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 681-690. ACM, 2006.
Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, and Amnon Ta-Shma
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs
For a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal.
We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:
- Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n).
- Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.
Algebraic constructions
coding theory
linear algebra
list-decoding
polynomial method
pseudorandomness
Theory of computation~Randomness, geometry and discrete structures
Theory of computation~Pseudorandomness and derandomization
Theory of computation~Computational complexity and cryptography
Theory of computation~Algebraic complexity theory
4:1-4:16
Regular Paper
https://eccc.weizmann.ac.il/report/2018/017/
Venkatesan
Guruswami
Venkatesan Guruswami
Department of Computer Science, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213
https://orcid.org/0000-0001-7926-3396
Research supported in part by NSF grants CCF-1422045 and CCF-1563742.
Nicolas
Resch
Nicolas Resch
Department of Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213
Research supported in part by NSF grants CCF-1618280, CCF-1422045, NSF CAREER award CCF-1750808 and NSERC grant CGSD2-502898.
Chaoping
Xing
Chaoping Xing
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371
https://orcid.org/0000-0002-1257-1033
10.4230/LIPIcs.CCC.2018.4
Boaz Barak, Russell Impagliazzo, Amir Shpilka, and Avi Wigderson. Personal Communication to Dvir-Shpilka [6], 2004.
Jean Bourgain. Expanders and dimensional expansion. Comptes Rendus Mathematique, 347(7-8):357-362, 2009. URL: http://dx.doi.org/10.1016/j.crma.2009.02.009.
http://dx.doi.org/10.1016/j.crma.2009.02.009
Jean Bourgain and Amir Yehudayoff. Expansion in SL₂(ℝ) and monotone expanders. Geometric and Functional Analysis, 23(1):1-41, 2013. Preliminary version in the 44th Annual ACM Symposium on Theory of Computing (STOC 2012). This work is the full version of [2]. URL: http://dx.doi.org/10.1007/s00039-012-0200-9.
http://dx.doi.org/10.1007/s00039-012-0200-9
Zeev Dvir and Shachar Lovett. Subspace evasive sets. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 351-358. ACM, 2012.
Zeev Dvir and Amir Shpilka. Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM J. Comput., 36(5):1404-1434, 2007. URL: http://dx.doi.org/10.1137/05063605X.
http://dx.doi.org/10.1137/05063605X
Zeev Dvir and Amir Shpilka. Towards dimension expanders over finite fields. Combinatorica, 31(3):305-320, 2011.
Zeev Dvir and Avi Wigderson. Monotone expanders: Constructions and applications. Theory of Computing, 6(1):291-308, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a012.
http://dx.doi.org/10.4086/toc.2010.v006a012
Michael A Forbes and Venkatesan Guruswami. Dimension expanders via rank condensers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 800-814, 2015.
Michael A Forbes and Amir Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 163-172. ACM, 2012.
Ernst M. Gabidulin. Theory of codes with maximum rank distance. Probl. Inform. Transm., 21(1):1-12, 1985. URL: http://www.mathnet.ru/eng/ppi967.
http://www.mathnet.ru/eng/ppi967
Ariel Gabizon. Deterministic extractors for affine sources over large fields. In Deterministic Extraction from Weak Random Sources, pages 33-53. Springer, 2011.
Venkatesan Guruswami. Linear-algebraic list decoding of folded reed-solomon codes. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, June 8-10, 2011, pages 77-85. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/CCC.2011.22.
http://dx.doi.org/10.1109/CCC.2011.22
Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. Combinatorica, 36(2):161-185, 2016.
Venkatesan Guruswami and Atri Rudra. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Trans. Information Theory, 54(1):135-150, 2008. URL: http://dx.doi.org/10.1109/TIT.2007.911222.
http://dx.doi.org/10.1109/TIT.2007.911222
Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM (JACM), 56(4):20, 2009.
Venkatesan Guruswami and Carol Wang. Linear-algebraic list decoding for variants of Reed-Solomon codes. IEEE Transactions on Information Theory, 59(6):3257-3268, 2013.
Venkatesan Guruswami and Carol Wang. Evading subspaces over large fields and explicit list-decodable rank-metric codes. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, volume 28 of LIPIcs, pages 748-761. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.748.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.748
Venkatesan Guruswami, Carol Wang, and Chaoping Xing. Explicit list-decodable rank-metric and subspace codes via subspace designs. IEEE Transactions on Information Theory, 62(5):2707-2718, 2016.
Venkatesan Guruswami and Chaoping Xing. Folded codes from function field towers and improved optimal rate list decoding. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 339-350. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214009.
http://dx.doi.org/10.1145/2213977.2214009
Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 843-852. ACM, 2013.
Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. Subspace designs based on algebraic function fields. Transactions of the AMS, 2017. To appear. Available as arXiv:1704.05992.
Aram W. Harrow. Quantum expanders from any classical cayley graph expander. Quantum Information & Computation, 8(8-9):715-721, 2008. URL: http://www.rintonpress.com/journals/qiconline.html#v8n89.
http://www.rintonpress.com/journals/qiconline.html#v8n89
Zohar S. Karnin and Amir Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333-364, 2011.
Alexander Lubotzky and Efim Zelmanov. Dimension expanders. Journal of Algebra, 319(2):730-738, 2008.
Farzad Parvaresh and Alexander Vardy. Correcting errors beyond the Guruswami-Sudan radius in polynomial time. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 285-294, 2005.
Pavel Pudlák and Vojtěch Rödl. Pseudorandom sets and explicit constructions of Ramsey graphs. In Complexity of computations and proofs, volume 13 of Quad. Mat., pages 327-346. Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics, 155(1):157-187, 2002.
Salil P. Vadhan. Pseudorandomness. Foundations and Trendsregistered in Theoretical Computer Science, 7(1-3):1-336, 2012.
Avi Wigderson. Expanders: Old and new applications and problems. Lecture at the Institute for Pure and Applied Mathematics (IPAM), 2004.
Venkatesan Guruswami, Nicolas Resch, and Chaoping Xing
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits
The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have demonstrated the central role of this problem and its variations in diverse areas such as cryptography, derandomization, proof complexity, learning theory, and circuit lower bounds.
The NP-hardness of computing the minimum numbers of terms in a DNF formula consistent with a given truth table was proved by W. Masek [William J. Masek, 1979] in 1979. In this work, we make the first progress in showing NP-hardness for more expressive classes of circuits, and establish an analogous result for the MCSP problem for depth-3 circuits of the form OR-AND-MOD_2. Our techniques extend to an NP-hardness result for MOD_m gates at the bottom layer under inputs from (Z / m Z)^n.
NP-hardness
Minimum Circuit Size Problem
depth-3 circuits
Theory of computation~Problems, reductions and completeness
5:1-5:31
Regular Paper
Shuichi
Hirahara
Shuichi Hirahara
Department of Computer Science, The University of Tokyo, Tokyo, Japan
Supported by ACT-I, JST and JSPS KAKENHI Grant Numbers JP16J06743
Igor C.
Oliveira
Igor C. Oliveira
Department of Computer Science, University of Oxford, Oxford, United Kingdom
Rahul
Santhanam
Rahul Santhanam
Department of Computer Science, University of Oxford, Oxford, United Kingdom
10.4230/LIPIcs.CCC.2018.5
Adi Akavia, Andrej Bogdanov, Siyao Guo, Akshay Kamath, and Alon Rosen. Candidate weak pseudorandom functions in AC^0 ∘ MOD_2. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 251-260. ACM, 2014. URL: http://dx.doi.org/10.1145/2554797.2554821.
http://dx.doi.org/10.1145/2554797.2554821
Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: http://dx.doi.org/10.1137/050628994.
http://dx.doi.org/10.1137/050628994
Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. 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 25-32. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_3.
http://dx.doi.org/10.1007/978-3-662-44465-8_3
Eric Allender, Joshua A. Grochow, and Cristopher Moore. Graph isomorphism and circuit size. CoRR, abs/1511.08189, 2015. URL: http://arxiv.org/abs/1511.08189.
http://arxiv.org/abs/1511.08189
Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing disjunctive normal form formulas and AC0 circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. URL: http://dblp.uni-trier.de/db/journals/siamcomp/siamcomp38.html#AllenderHMPS08.
http://dblp.uni-trier.de/db/journals/siamcomp/siamcomp38.html#AllenderHMPS08
Eric Allender and Shuichi Hirahara. New insights on the (non-)hardness of circuit minimization and related problems. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, volume 83 of LIPIcs, pages 54:1-54:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2017.54.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2017.54
Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 21-33. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21
Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci., 77(1):14-40, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.004.
http://dx.doi.org/10.1016/j.jcss.2010.06.004
Yossi Azar, Rajeev Motwani, and Joseph Naor. Approximating probability distributions using small sample spaces. Combinatorica, 18(2):151-171, 1998. URL: http://dx.doi.org/10.1007/PL00009813.
http://dx.doi.org/10.1007/PL00009813
Andrej Bogdanov and Alon Rosen. Pseudorandom functions: Three decades later. In Yehuda Lindell, editor, Tutorials on the Foundations of Cryptography., pages 79-158. Springer International Publishing, 2017. URL: http://dx.doi.org/10.1007/978-3-319-57048-8_3.
http://dx.doi.org/10.1007/978-3-319-57048-8_3
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 10:1-10:24. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10
Arkadev Chattopadhyay and Shachar Lovett. Linear systems over finite abelian groups. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, June 8-10, 2011, pages 300-308. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/CCC.2011.25.
http://dx.doi.org/10.1109/CCC.2011.25
Arkadev Chattopadhyay and Avi Wigderson. Linear systems over composite moduli. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 43-52. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.17.
http://dx.doi.org/10.1109/FOCS.2009.17
Gil Cohen and Igor Shinkar. The complexity of DNF of parities. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 47-58. ACM, 2016. URL: http://dx.doi.org/10.1145/2840728.2840734.
http://dx.doi.org/10.1145/2840728.2840734
Sebastian Czort. The complexity of minimizing disjunctive normal form formulas. Master’s Thesis, University of Aarhus, 1999.
Anindya De, Omid Etesami, Luca Trevisan, and Madhur Tulsiani. Improved pseudorandom generators for depth 2 circuits. Electronic Colloquium on Computational Complexity (ECCC), 16:141, 2009. URL: http://eccc.hpi-web.de/report/2009/141.
http://eccc.hpi-web.de/report/2009/141
Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998.
Vitaly Feldman. Hardness of approximate two-level logic minimization and PAC learning with membership queries. J. Comput. Syst. Sci., 75(1):13-26, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2008.07.007.
http://dx.doi.org/10.1016/j.jcss.2008.07.007
M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
Chris Godsil. Double orthogonal complement of a finite module. MathOverflow (Retrieved 19-01-2018). URL: https://mathoverflow.net/q/75268.
https://mathoverflow.net/q/75268
Parikshit Gopalan, Daniel M. Kane, and Raghu Meka. Pseudorandomness via the discrete fourier transform. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 903-922. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.60.
http://dx.doi.org/10.1109/FOCS.2015.60
Vince Grolmusz. A lower bound for depth-3 circuits with MOD m gates. Inf. Process. Lett., 67(2):87-90, 1998. URL: http://dx.doi.org/10.1016/S0020-0190(98)00093-3.
http://dx.doi.org/10.1016/S0020-0190(98)00093-3
Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18
John M. Hitchcock and Aduri Pavan. On the np-completeness of the minimum circuit size problem. In Prahladh Harsha and G. Ramalingam, editors, 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, volume 45 of LIPIcs, pages 236-245. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.236.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.236
Stasys Jukna. On graph complexity. Combinatorics, Probability & Computing, 15(6):855-876, 2006. URL: http://dx.doi.org/10.1017/S0963548306007620.
http://dx.doi.org/10.1017/S0963548306007620
Stasys Jukna. Boolean Function Complexity - Advances and Frontiers. Springer, 2012.
Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 73-79. ACM, 2000. URL: http://dx.doi.org/10.1145/335305.335314.
http://dx.doi.org/10.1145/335305.335314
Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York., The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: http://www.cs.berkeley.edu/~luca/cs172/karp.pdf.
http://www.cs.berkeley.edu/~luca/cs172/karp.pdf
Subhash Khot and Rishi Saket. Hardness of minimizing and learning DNF expressions. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 231-240. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.37.
http://dx.doi.org/10.1109/FOCS.2008.37
Jan Krajícek. Forcing with Random Variables and Proof Complexity, volume 382 of London Mathematical Society lecture note series. Cambridge University Press, 2011. URL: http://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/forcing-random-variables-and-proof-complexity?format=PB.
http://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/forcing-random-variables-and-proof-complexity?format=PB
William J. Masek. Some NP-complete set covering problems. Unpublished Manuscript, 1979.
Cody D. Murray and Richard Ryan Williams. On the (non) np-hardness of computing circuit complexity. In David Zuckerman, editor, 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, volume 33 of LIPIcs, pages 365-380. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput., 22(4):838-856, 1993. URL: http://dx.doi.org/10.1137/0222053.
http://dx.doi.org/10.1137/0222053
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput., 22(4):838-856, 1993. URL: http://dblp.uni-trier.de/db/journals/siamcomp/siamcomp22.html#NaorN93.
http://dblp.uni-trier.de/db/journals/siamcomp/siamcomp22.html#NaorN93
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions
Igor Carboni Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 18:1-18:49. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.18.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.18
Ramamohan Paturi, Michael E. Saks, and Francis Zane. Exponential lower bounds for depth three boolean circuits. Computational Complexity, 9(1):1-15, 2000. URL: http://dx.doi.org/10.1007/PL00001598.
http://dx.doi.org/10.1007/PL00001598
Leonard Pitt and Leslie G. Valiant. Computational limitations on learning from examples. J. ACM, 35(4):965-984, 1988. URL: http://dx.doi.org/10.1145/48014.63140.
http://dx.doi.org/10.1145/48014.63140
Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997. URL: http://dx.doi.org/10.1006/jcss.1997.1494.
http://dx.doi.org/10.1006/jcss.1997.1494
Petr Slavík. A tight analysis of the greedy algorithm for set cover. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 435-441. ACM, 1996. URL: http://dx.doi.org/10.1145/237814.237991.
http://dx.doi.org/10.1145/237814.237991
Boris A. Trakhtenbrot. A survey of russian approaches to perebor (brute-force searches) algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984. URL: http://dx.doi.org/10.1109/MAHC.1984.10036.
http://dx.doi.org/10.1109/MAHC.1984.10036
Luca Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 453-461. ACM, 2001. URL: http://dx.doi.org/10.1145/380752.380839.
http://dx.doi.org/10.1145/380752.380839
Shuichi Hirahara, Igor C. Oliveira, and Rahul Santhanam
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials
We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over R) of functions from some "simple" class C. In particular, given C we are interested in finding low-complexity functions lacking sparse representations. When C forms a basis for the space of Boolean functions (e.g., the set of PARITY functions or the set of conjunctions) this sort of problem has a well-understood answer; the problem becomes interesting when C is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where C is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts.
We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. Let alpha(n) be an unbounded function such that n^{alpha(n)} is time constructible (e.g. alpha(n) = log^*(n)). We show:
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many linear threshold functions to represent (depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds).
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many ReLU gates to represent (depth-two neural networks with ReLU activation function).
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many O(1)-degree F_p-polynomials to represent exactly, for every prime p (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in E^{NP} requiring 2^{Omega(n)} linear combinations.
- Functions in NTIME[n^{poly(log n)}] that require super-polynomially many ACC ° THR circuits to represent exactly (further generalizing the recent lower bounds of Murray and the author). We also obtain "fixed-polynomial" lower bounds for functions in NP, for the first three representation classes. All our lower bounds are obtained via algorithms for analyzing linear combinations of simple functions in the above scenarios, in ways which substantially beat exhaustive search.
linear threshold functions
lower bounds
neural networks
low-degree polynomials
Theory of computation~Circuit complexity
Computer systems organization~Neural networks
6:1-6:24
Regular Paper
Richard Ryan
Williams
Richard Ryan Williams
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
https://orcid.org/0000-0003-2326-2233
Supported by NSF CCF-1553288. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
10.4230/LIPIcs.CCC.2018.6
Amir Abboud, Aviad Rubinstein, and R. Ryan Williams. Distributed PCP theorems for hardness of approximation in P. In FOCS, pages 25-36, 2017.
Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In FOCS, pages 467-476, 2016.
Raman Arora, Amitabh Basu, Poorya Mianjy, and Anirbit Mukherjee. Understanding deep neural networks with rectified linear units. arXiv preprint arXiv:1611.01491, 2016.
László Babai, Kristoffer Arnsfelt Hansen, Vladimir V. Podolskii, and Xiaoming Sun. Weights of exact threshold functions. In Mathematical Foundations of Computer Science, pages 66-77, 2010.
David A. Mix Barrington, Howard Straubing, and Denis Thérien. Non-uniform automata over groups. Inf. Comput., 89(2):109-132, 1990.
Richard Beigel and Jun Tarui. On ACC. Computational Complexity, pages 350-366, 1994.
Eli Ben-Sasson and Emanuele Viola. Short PCPs with projection queries. In ICALP, pages 163-173, 2014.
Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009.
Jean Bourgain. Estimation of certain exponential sums arising in complexity theory. C.R. Acad. Sci. Paris Ser. I, 340:627-631, 2005.
Jin-yi Cai, Frederic Green, and Thomas Thierauf. On the correlation of symmetric functions. Mathematical Systems Theory, 29(3):245-258, 1996.
Chris Calabro. A lower bound on the size of series-parallel graphs dense in long paths. Electronic Colloquium on Computational Complexity (ECCC), 15(110), 2008.
Timothy M. Chan and Ryan Williams. Deterministic APSP, Orthogonal Vectors, and more: Quickly derandomizing Razborov-Smolensky. In SODA, pages 1246-1255, 2016.
Arkadev Chattopadhyay and Nikhil S. Mande. Weights at the bottom matter when the top is heavy. Electronic Colloquium on Computational Complexity (ECCC), 24:83, 2017.
Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan. Average-case lower bounds and satisfiability algorithms for small threshold circuits. In CCC, pages 1:1-1:35, 2016.
Amit Daniely. Depth separation for neural networks. In Proceedings of COLT, pages 690-696, 2017.
Ronen Eldan and Ohad Shamir. The power of depth for feedforward neural networks. In Proceedings of COLT, pages 907-940, 2016.
Yuval Filmus, Hamed Hatami, Steven Heilman, Elchanan Mossel, Ryan O'Donnell, Sushant Sachdeva, Andrew Wan, and Karl Wimmer. Real Analysis in Computer Science: A collection of open problems, Simons Institute, 2014. URL: https://simons.berkeley.edu/sites/default/files/openprobsmerged.pdf.
https://simons.berkeley.edu/sites/default/files/openprobsmerged.pdf
Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Springer, 2010.
Anna Gál and Vladimir Trifonov. On the correlation between parity and modular polynomials. Theory Comput. Syst., 50(3):516-536, 2012.
Frederic Green. The correlation between parity and quadratic polynomials mod 3. Journal of Computer and System Sciences, 69(1):28-44, 2004.
Jacques Hadamard. Résolution d'une question relative aux déterminants. Bull. Sci. Math., 17:30-31, 1893.
András Hajnal, Wolfgang Maass, Pavel Pudlák, Mario Szegedy, and György Turán. Threshold circuits of bounded depth. J. Comput. Syst. Sci., 46(2):129-154, 1993.
Kristoffer Arnsfelt Hansen and Vladimir V Podolskii. Exact threshold circuits. In CCC, pages 270-279, 2010.
Johan Håstad and Mikael Goldmann. On the power of small-depth threshold circuits. Computational Complexity, 1:113-129, 1991.
Hamed Hatami, Pooya Hatami, and Shachar Lovett. Higher-order Fourier analysis and applications. Manuscript, 2016. URL: https://cseweb.ucsd.edu/~slovett/files/survey-higher_order_fourier.pdf.
https://cseweb.ucsd.edu/~slovett/files/survey-higher_order_fourier.pdf
Ellis Horowitz and Sartaj Sahni. Computing partitions with applications to the knapsack problem. JACM, 21(2):277-292, 1974.
Hamid Jahanjou, Eric Miles, and Emanuele Viola. Local reductions. In Proceedings of ICALP, pages 749-760, 2015.
Daniel M. Kane and Ryan Williams. Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuits. In STOC, pages 633-643, 2016.
Daniel Lokshtanov, Ramamohan Paturi, Suguru Tamaki, R. Ryan Williams, and Huacheng Yu. Beating brute force for systems of polynomial equations over finite fields. In SODA, pages 2190-2202, 2017.
Shachar Lovett. Personal communication, 2017.
Wolfgang Maass. Bounds for the computational power and learning complexity of analog neural nets. SIAM Journal on Computing, 26(3):708-732, 1997.
Peter Bro Miltersen, N. V. Vinodchandran, and Osamu Watanabe. Super-polynomial versus half-exponential circuit size in the exponential hierarchy. In COCOON, Springer LNCS 1627, pages 210-220, 1999.
Anirbit Mukherjee and Amitabh Basu. Lower bounds over Boolean inputs for deep neural networks with ReLU gates. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1711.03073.
http://arxiv.org/abs/1711.03073
S. Muroga, I. Toda, and S. Takasu. Theory of majority decision elements. Journal of the Franklin Institute, 271:376-418, 1961.
Cody Murray and Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime: An easy witness lemma for NP and NQP. Electronic Colloquium on Computational Complexity (ECCC), TR17-188, 2017.
Noam Nisan. The communication complexity of threshold gates. In Proceedings of "Combinatorics, Paul Erdos is Eighty", pages 301-315, 1994.
Mihai Pǎtraşcu and Ryan Williams. On the possibility of faster sat algorithms. In SODA, pages 1065-1075, 2010.
Vwani P. Roychowdhury, Alon Orlitsky, and Kai-Yeung Siu. Lower bounds on threshold and related circuits via communication complexity. IEEE Transactions on Information Theory, 40(2):467-474, 1994.
Itay Safran and Ohad Shamir. Depth-width tradeoffs in approximating natural functions with neural networks. In International Conference on Machine Learning, pages 2979-2987, 2017.
Rahul Santhanam. Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009.
Rahul Santhanam and Ryan Williams. On medium-uniformity and circuit lower bounds. In IEEE Conf. Computational Complexity, pages 15-23, 2013.
Joel Seiferas, Michael Fischer, and Albert Meyer. Separating nondeterministic time complexity classes. Journal of the ACM, 25(1):146-167, jan 1978.
Suguru Tamaki. A satisfiability algorithm for depth two circuits with a sub-quadratic number of symmetric and threshold gates. Electronic Colloquium on Computational Complexity (ECCC), 23:100, 2016.
Matus Telgarsky. benefits of depth in neural networks. In Proceedings of COLT, pages 1517-1539, 2016.
Roei Tell. Proving that prBPP=prP is as hard as "almost" proving that P ≠ NP. Electronic Colloquium on Computational Complexity (ECCC), 18(3), 2018.
S. Toda. PP is as hard as the polynomial-time hierarchy. sicomp, 20(5):865-877, 1991.
L. G. Valiant. Graph-theoretic arguments in low-level complexity. In J. Gruska, editor, MFCS, volume 53 of LNCS, pages 162-176, Tatranská Lomnica, Czechoslovakia, sep 1977. Springer.
Virginia Vassilevska Williams and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. SIAM Journal on Computing, 42(3):831-854, 2013.
Emanuele Viola. Guest column: correlation bounds for polynomials over 0, 1. SIGACT News, 40(1):27-44, 2009.
Ryan Williams. A casual tour around a circuit complexity bound. SIGACT News, 42(3):54-76, 2011.
Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. sicomp, 42(3):1218-1244, 2013.
Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. In STOC, pages 194-202, 2014.
Ryan Williams. Nonuniform ACC circuit lower bounds. JACM, 61(1):2, 2014.
Ryan Williams. Counting solutions to polynomial systems via reductions. In Raimund Seidel, editor, 1st Symposium on Simplicity in Algorithms (SOSA 2018), pages 6:1-6:15, 2018.
R. O. Winder. Threshold Logic. PhD thesis, Princeton University, 1962. Preliminary version in FOCS'60.
Stanislav Žák. A Turing machine time hierarchy. Theoretical Computer Science, 26(3):327-333, 1983.
Richard Ryan Williams
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Power of Natural Properties as Oracles
We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions.
natural properties
Minimal Circuit Size Problem (MCSP)
circuit lower bounds
hardness of MCSP
learning algorithms
obfuscation
Indistinguishability Obfuscators (IO)
Theory of computation~Computational complexity and cryptography
7:1-7:20
Regular Paper
Russell
Impagliazzo
Russell Impagliazzo
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Valentine
Kabanets
Valentine Kabanets
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Ilya
Volkovich
Ilya Volkovich
Department of EECS, CSE Division, University of Michigan, Ann Arbor, MI, USA
10.4230/LIPIcs.CCC.2018.7
L. M. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 75-83, 1978.
Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: http://dx.doi.org/10.1137/050628994.
http://dx.doi.org/10.1137/050628994
Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. 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 25-32. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_3.
http://dx.doi.org/10.1007/978-3-662-44465-8_3
Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing disjunctive normal form formulas and ac^0 circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. URL: http://dx.doi.org/10.1137/060664537.
http://dx.doi.org/10.1137/060664537
Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 21-33. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21
S. Arora and B. Barak. Computational complexity: a modern approach. Cambridge University Press, 2009.
Vikraman Arvind and Johannes Köbler. On pseudorandomness and resource-bounded measure. Theor. Comput. Sci., 255(1-2):205-221, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(99)00164-4.
http://dx.doi.org/10.1016/S0304-3975(99)00164-4
L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991.
L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993.
B. Barak. A probabilistic-time hierarchy theorem for "slightly non-uniform" algorithms. In RANDOM, pages 194-208, 2002.
B. Barak, O. Goldreich, R. Impagliazzo, S. Rudich, A. Sahai, S. P. Vadhan, and K. Yang. On the (im)possibility of obfuscating programs. In Advances in Cryptology - CRYPTO 2001, 21st Annual International Cryptology Conference, pages 1-18, 2001.
Donald Beaver and Joan Feigenbaum. Hiding instances in multioracle queries. In Christian Choffrut and Thomas Lengauer, editors, STACS 90, 7th Annual Symposium on Theoretical Aspects of Computer Science, Rouen, France, February 22-24, 1990, Proceedings, volume 415 of Lecture Notes in Computer Science, pages 37-48. Springer, 1990. URL: http://dx.doi.org/10.1007/3-540-52282-4_30.
http://dx.doi.org/10.1007/3-540-52282-4_30
Nader H. Bshouty, Richard Cleve, Ricard Gavaldà, Sampath Kannan, and Christino Tamon. Oracles and queries that are sufficient for exact learning. J. Comput. Syst. Sci., 52(3):421-433, 1996. URL: http://dx.doi.org/10.1006/jcss.1996.0032.
http://dx.doi.org/10.1006/jcss.1996.0032
H. Buhrman, L. Fortnow, and T. Thierauf. Nonrelativizing separations. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity (CCC), pages 8-12, 1998.
Harry Buhrman and Steven Homer. Superpolynomial circuits, almost sparse oracles and the exponential hierarchy. In R. K. Shyamasundar, editor, Foundations of Software Technology and Theoretical Computer Science, 12th Conference, New Delhi, India, December 18-20, 1992, Proceedings, volume 652 of Lecture Notes in Computer Science, pages 116-127. Springer, 1992. URL: http://dx.doi.org/10.1007/3-540-56287-7_99.
http://dx.doi.org/10.1007/3-540-56287-7_99
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 10:1-10:24. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10
J. Feigenbaum and L. Fortnow. Random-self-reducibility of complete sets. SIAM J. on Computing, 22(5):994-1005, 1993.
L. Fortnow and A. R. Klivans. Efficient learning algorithms yield circuit lower bounds. J. Comput. Syst. Sci., 75(1):27-36, 2009.
L. Fortnow and R. Santhanam. Hierarchy theorems for probabilistic polynomial time. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 316-324, 2004.
O. Goldreich and D. Zuckerman. Another proof that BPP ⊆ PH (and more). Studies in Complexity and Cryptography, pages 40-53, 2011.
S. Goldwasser and G. N. Rothblum. On best-possible obfuscation. In Theory of Cryptography, 4th Theory of Cryptography Conference, TCC, pages 194-213, 2007.
Hans Heller. On relativized exponential and probabilistic complexity classes. Information and Control, 71(3):231-243, 1986. URL: http://dx.doi.org/10.1016/S0019-9958(86)80012-2.
http://dx.doi.org/10.1016/S0019-9958(86)80012-2
Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18
John M. Hitchcock and Aduri Pavan. On the np-completeness of the minimum circuit size problem. In Prahladh Harsha and G. Ramalingam, editors, 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, volume 45 of LIPIcs, pages 236-245. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.236.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.236
R. Impagliazzo, V. Kabanets, and A. Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. J. of Computer and System Sciences, 65(4):672-694, 2002.
R. Impagliazzo and A. Wigderson. P=BPP unless E has subexponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pages 220-229, 1997.
R. Impagliazzo and A. Wigderson. Randomness vs. time: De-randomization under a uniform assumption. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 734-743, 1998.
V. Kabanets and J.-Y. Cai. Circuit minimization problem. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 73-79, 2000.
Ravi Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55(1-3):40-56, 1982. URL: http://dx.doi.org/10.1016/S0019-9958(82)90382-5.
http://dx.doi.org/10.1016/S0019-9958(82)90382-5
Richard M. Karp. Turing award lecture. In Bill Healy and Judith D. Schlesinger, editors, Proceedings of the 1985 ACM annual conference on The range of computing: mid-80’s perspective: mid-80’s perspective, Denver, Colorado, USA, October 14-16, 1985, page 193. ACM, 1985. URL: http://dx.doi.org/10.1145/320435.320497.
http://dx.doi.org/10.1145/320435.320497
Richard M. Karp and Richard J. Lipton. Some connections between nonuniform and uniform complexity classes. In Raymond E. Miller, Seymour Ginsburg, Walter A. Burkhard, and Richard J. Lipton, editors, Proceedings of the 12th Annual ACM Symposium on Theory of Computing, April 28-30, 1980, Los Angeles, California, USA, pages 302-309. ACM, 1980. URL: http://dx.doi.org/10.1145/800141.804678.
http://dx.doi.org/10.1145/800141.804678
A. Klivans, P. Kothari, and I. Oliveira. Constructing hard functions from learning algorithms. In Proceedings of the 28th Annual IEEE Conference on Computational Complexity (CCC), pages 86-97, 2013.
Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL: http://dx.doi.org/10.1137/S0097539700389652.
http://dx.doi.org/10.1137/S0097539700389652
Johannes Köbler and Osamu Watanabe. New collapse consequences of NP having small circuits. SIAM J. Comput., 28(1):311-324, 1998. URL: http://dx.doi.org/10.1137/S0097539795296206.
http://dx.doi.org/10.1137/S0097539795296206
Ilan Komargodski, Tal Moran, Moni Naor, Rafael Pass, Alon Rosen, and Eylon Yogev. One-way functions and (im)perfect obfuscation. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 374-383. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.47.
http://dx.doi.org/10.1109/FOCS.2014.47
C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. JACM, 39(4):859-868, 1992.
D. van Melkebeek and K. Pervyshev. A generic time hierarchy with one bit of advice. Computational Complexity, 16(2):139-179, 2007.
Cody D. Murray and Richard Ryan Williams. On the (non) np-hardness of computing circuit complexity. In David Zuckerman, editor, 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, volume 33 of LIPIcs, pages 365-380. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365
N. Nisan and A. Wigderson. Hardness vs. randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994.
I. C. Oliveira and R. Santhanam. Conspiracies between learning algorithms, circuit lower bounds and pseudorandomness. CoRR, abs/1611.01190, 2016. URL: http://arxiv.org/abs/1611.01190.
http://arxiv.org/abs/1611.01190
A. A. Razborov and S. Rudich. Natural proofs. J. of Computer and System Sciences, 55(1):24-35, 1997.
R. Santhanam. Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009.
S. Toda. PP is as hard as the polynomial time hierarchy. SIAM J. on Computing, 20(5):865-877, 1991.
Boris A. Trakhtenbrot. A survey of russian approaches to perebor (brute-force searches) algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984. URL: http://dx.doi.org/10.1109/MAHC.1984.10036.
http://dx.doi.org/10.1109/MAHC.1984.10036
L. Trevisan and S. P. Vadhan. Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity, 16(4):331-364, 2007.
C. Umans. Pseudo-random generators for all hardnesses. J. of Computer and System Sciences, 67(2):419-440, 2003.
L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979.
L. G. Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, 1984.
I. Volkovich. On learning, lower bounds and (un)keeping promises. In Proceedings of the 41st ICALP, pages 1027-1038, 2014.
R. Impagliazzo, V. Kabanets, and I. Volkovich
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Linear Sketching over F_2
We initiate a systematic study of linear sketching over F_2. For a given Boolean function treated as f : F_2^n -> F_2 a randomized F_2-sketch is a distribution M over d x n matrices with elements over F_2 such that Mx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms.
Motivated by these applications we study a connection between F_2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F_2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F_2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F_2-sketching is optimal (up to constant factors) for uniformly distributed inputs.
Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F_2 can be constructed as F_2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates.
Linear sketch
Streaming algorithms
XOR-functions
Communication complexity
Theory of computation~Communication complexity
8:1-8:37
Regular Paper
https://arxiv.org/pdf/1611.01879.pdf
Sampath
Kannan
Sampath Kannan
University of Pennsylvania
This work was supported by grants NSF CICI 1547360 and ONR N00014-15-1-2006.
Elchanan
Mossel
Elchanan Mossel
Massachusetts Institute of Technology
E.M. acknowledges the support of grant N00014-16-1-2227 from Office of Naval Research and of NSF awards CCF 1320105 and DMS-1737944 as well as support from Simons Think Tank on Geometry & Algorithms.
Swagato
Sanyal
Swagato Sanyal
Division of Mathematical Sciences, Nanyang Technological University, Singapore and Centre for Quantum Technologies, National University of Singapore, Singapore
S.S. acknowledges the support by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. Part of this work was done when S. S. was a visiting research fellow at the Tata Institute of Fundamental Research, Mumbai.
Grigory
Yaroslavtsev
Grigory Yaroslavtsev
Indiana University, Bloomington
This work was supported by NSF award 1657477.
10.4230/LIPIcs.CCC.2018.8
Yuqing Ai, Wei Hu, Yi Li, and David P. Woodruff. New characterizations in turnstile streams with applications. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 20:1-20:22. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.20.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.20
Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing reed-muller codes. IEEE Trans. Information Theory, 51(11):4032-4039, 2005.
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. Syst. Sci., 58(1):137-147, 1999. URL: http://dx.doi.org/10.1006/jcss.1997.1545.
http://dx.doi.org/10.1006/jcss.1997.1545
Sepehr Assadi, Sanjeev Khanna, and Yang Li. On estimating maximum matching size in graph streams. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1723-1742, 2017.
Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1345-1364, 2016.
Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of reed-muller codes. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 488-497, 2010.
Eric Blais, Li-Yang Tan, and Andrew Wan. An inequality for the fourier spectrum of parity decision trees. CoRR, abs/1506.01055, 2015. URL: http://arxiv.org/abs/1506.01055.
http://arxiv.org/abs/1506.01055
Andrej Bogdanov and Emanuele Viola. Pseudorandom bits for polynomials. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 41-51, 2007.
Anirban Dasgupta, Ravi Kumar, and D. Sivakumar. Sparse and lopsided set disjointness via information theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, pages 517-528, 2012.
Sumit Ganguly. Lower bounds on frequency estimation of data streams (extended abstract). In Computer Science - Theory and Applications, Third International Computer Science Symposium in Russia, CSR 2008, Moscow, Russia, June 7-12, 2008, Proceedings, pages 204-215, 2008.
Dmitry Gavinsky, Julia Kempe, and Ronald de Wolf. Quantum communication cannot simulate a public coin. CoRR, quant-ph/0411051, 2004. URL: http://arxiv.org/abs/quant-ph/0411051.
http://arxiv.org/abs/quant-ph/0411051
Mohsen Ghaffari and Merav Parter. MST in log-star rounds of congested clique. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 19-28, 2016.
Mika Göös and T. S. Jayram. A composition theorem for conical juntas. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 5:1-5:16, 2016.
Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 257-266, 2015.
Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1077-1088, 2015.
Parikshit Gopalan, Ryan O'Donnell, Rocco A. Servedio, Amir Shpilka, and Karl Wimmer. Testing fourier dimensionality and sparsity. SIAM J. Comput., 40(4):1075-1100, 2011. URL: http://dx.doi.org/10.1137/100785429.
http://dx.doi.org/10.1137/100785429
Vince Grolmusz. On the power of circuits with gates of low l_1 norms. Theor. Comput. Sci., 188(1-2):117-128, 1997. URL: http://dx.doi.org/10.1016/S0304-3975(96)00290-3.
http://dx.doi.org/10.1016/S0304-3975(96)00290-3
Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 282-288, 2016.
James W. Hegeman, Gopal Pandurangan, Sriram V. Pemmaraju, Vivek B. Sardeshmukh, and Michele Scquizzato. Toward optimal bounds in the congested clique: Graph connectivity and MST. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, pages 91-100, 2015.
Wei Huang, Yaoyun Shi, Shengyu Zhang, and Yufan Zhu. The communication complexity of the hamming distance problem. Inf. Process. Lett., 99(4):149-153, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.01.014.
http://dx.doi.org/10.1016/j.ipl.2006.01.014
T. S. Jayram. Information complexity: a tutorial. In Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2010, June 6-11, 2010, Indianapolis, Indiana, USA, pages 159-168, 2010.
T. S. Jayram, Ravi Kumar, and D. Sivakumar. Two applications of information complexity. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 673-682, 2003.
Tomasz Jurdzinski and Krzysztof Nowicki. MST in O(1) rounds of congested clique. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2620-2632, 2018.
Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570, 2014.
Michael Kapralov, Jelani Nelson, Jakub Pachocki, Zhengyu Wang, David P. Woodruff, and Mobin Yahyazadeh. Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 475-486, 2017.
Eyal Kushilevitz and Yishay Mansour. Learning decision trees using the fourier spectrum. SIAM J. Comput., 22(6):1331-1348, 1993. URL: http://dx.doi.org/10.1137/0222080.
http://dx.doi.org/10.1137/0222080
Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997.
Troy Lee and Shengyu Zhang. Composition theorems in communication complexity. In Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, pages 475-489, 2010.
Nikos Leonardos. An improved lower bound for the randomized decision tree complexity of recursive majority,. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 696-708, 2013.
Ming Lam Leung, Yang Li, and Shengyu Zhang. Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models. CoRR, abs/1101.4555, 2011. URL: http://arxiv.org/abs/1101.4555.
http://arxiv.org/abs/1101.4555
Yi Li, Huy L. Nguyen, and David P. Woodruff. Turnstile streaming algorithms might as well be linear sketches. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 174-183, 2014.
Yang Liu and Shengyu Zhang. Quantum and randomized communication complexity of XOR functions in the SMP model. Electronic Colloquium on Computational Complexity (ECCC), 20:10, 2013. URL: http://eccc.hpi-web.de/report/2013/010.
http://eccc.hpi-web.de/report/2013/010
Shachar Lovett. Unconditional pseudorandom generators for low degree polynomials. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 557-562, 2008.
Shachar Lovett. Recent advances on the log-rank conjecture in communication complexity. Bulletin of the EATCS, 112, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/260.
http://eatcs.org/beatcs/index.php/beatcs/article/view/260
Frédéric Magniez, Ashwin Nayak, Miklos Santha, Jonah Sherman, Gábor Tardos, and David Xiao. Improved bounds for the randomized decision tree complexity of recursive majority. CoRR, abs/1309.7565, 2013. URL: http://arxiv.org/abs/1309.7565.
http://arxiv.org/abs/1309.7565
Frédéric Magniez, Ashwin Nayak, Miklos Santha, and David Xiao. Improved bounds for the randomized decision tree complexity of recursive majority. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, pages 317-329, 2011.
Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Record, 43(1):9-20, 2014. URL: http://dx.doi.org/10.1145/2627692.2627694.
http://dx.doi.org/10.1145/2627692.2627694
Ashley Montanaro and Tobias Osborne. On the communication complexity of XOR functions. CoRR, abs/0909.3392, 2009. URL: http://arxiv.org/abs/0909.3392.
http://arxiv.org/abs/0909.3392
Elchanan Mossel, Sampath Kannan, and Grigory Yaroslavtsev. Linear sketching over 𝕗_2. Electronic Colloquium on Computational Complexity (ECCC), 23:174, 2016. URL: http://eccc.hpi-web.de/report/2016/174.
http://eccc.hpi-web.de/report/2016/174
Elchanan Mossel, Ryan O'Donnell, and Rocco A. Servedio. Learning juntas. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 206-212, 2003.
Ryan O'Donnell, John Wright, Yu Zhao, Xiaorui Sun, and Li-Yang Tan. A composition theorem for parity kill number. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 144-154, 2014.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996.
Michael E. Saks and Avi Wigderson. Probabilistic boolean decision trees and the complexity of evaluating game trees. In 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, 27-29 October 1986, pages 29-38, 1986.
Swagato Sanyal. Near-optimal upper bound on fourier dimension of boolean functions in terms of fourier sparsity. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 1035-1045, 2015.
Yaoyun Shi and Zhiqiang Zhang. Communication complexities of symmetric xor functions. Quantum Inf. Comput, pages 0808-1762, 2008.
Amir Shpilka, Avishay Tal, and Ben lee Volk. On the structure of boolean functions with small spectral norm. In Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 37-48, 2014.
Xiaoming Sun and Chengu Wang. Randomized communication complexity for linear algebra problems over finite fields. In 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, pages 477-488, 2012.
Justin Thaler. Semi-streaming algorithms for annotated graph streams. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 59:1-59:14, 2016.
Hing Yin Tsang, Chung Hoi Wong, Ning Xie, and Shengyu Zhang. Fourier sparsity, spectral norm, and the log-rank conjecture. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 658-667, 2013.
Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC 2008, 23-26 June 2008, College Park, Maryland, USA, pages 124-127, 2008.
Omri Weinstein and David P. Woodruff. The simultaneous communication of disjointness with applications to data streams. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 1082-1093, 2015.
David P. Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2):1-157, 2014. URL: http://dx.doi.org/10.1561/0400000060.
http://dx.doi.org/10.1561/0400000060
Andrew Chi-Chih Yao. Lower bounds by probabilistic arguments (extended abstract). In 24th Annual Symposium on Foundations of Computer Science, Tucson, Arizona, USA, 7-9 November 1983, pages 420-428, 1983.
Grigory Yaroslavtsev. Approximate linear sketching over 𝕗_2, 2017.
Zhiqiang Zhang and Yaoyun Shi. On the parity complexity measures of boolean functions. Theor. Comput. Sci., 411(26-28):2612-2618, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.03.027.
http://dx.doi.org/10.1016/j.tcs.2010.03.027
Sampath Kannan, Elchanan Mossel, Swagato Sanyal, and Grigory Yaroslavtsev
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Communication Complexity with Small Advantage
We study problems in randomized communication complexity when the protocol is only required to attain some small advantage over purely random guessing, i.e., it produces the correct output with probability at least epsilon greater than one over the codomain size of the function. Previously, Braverman and Moitra (STOC 2013) showed that the set-intersection function requires Theta(epsilon n) communication to achieve advantage epsilon. Building on this, we prove the same bound for several variants of set-intersection: (1) the classic "tribes" function obtained by composing with And (provided 1/epsilon is at most the width of the And), and (2) the variant where the sets are uniquely intersecting and the goal is to determine partial information about (say, certain bits of the index of) the intersecting coordinate.
Communication
complexity
small
advantage
Theory of computation~Communication complexity
9:1-9:17
Regular Paper
Supported by NSF grant CCF-1657377.
https://eccc.weizmann.ac.il/report/2016/148/
Thomas
Watson
Thomas Watson
Department of Computer Science, University of Memphis, Memphis, TN, USA
Supported by NSF grant CCF-1657377.
10.4230/LIPIcs.CCC.2018.9
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702-732, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.006.
http://dx.doi.org/10.1016/j.jcss.2003.11.006
Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 151-160. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488628.
http://dx.doi.org/10.1145/2488608.2488628
Mark Braverman and Ankur Moitra. An information complexity approach to extended formulations. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 161-170. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488629.
http://dx.doi.org/10.1145/2488608.2488629
Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Trans. Information Theory, 60(10):6058-6069, 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2347282.
http://dx.doi.org/10.1109/TIT.2014.2347282
Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Beyond set disjointness: the communication complexity of finding the intersection. In Magnús M. Halldórsson and Shlomi Dolev, editors, ACM Symposium on Principles of Distributed Computing, PODC '14, Paris, France, July 15-18, 2014, pages 106-113. ACM, 2014. URL: http://dx.doi.org/10.1145/2611462.2611501.
http://dx.doi.org/10.1145/2611462.2611501
Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Certifying equality with limited interaction. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, volume 28 of LIPIcs, pages 545-581. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.545.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.545
Amit Chakrabarti and Oded Regev. An optimal lower bound on the communication complexity of gap-hamming-distance. SIAM J. Comput., 41(5):1299-1317, 2012. URL: http://dx.doi.org/10.1137/120861072.
http://dx.doi.org/10.1137/120861072
Arkadev Chattopadhyay and Sagnik Mukhopadhyay. Tribes is hard in the message passing model. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 224-237. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.224.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.224
Mika Göös and T. S. Jayram. A composition theorem for conical juntas. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 5:1-5:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.5.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.5
Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 257-266. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746596.
http://dx.doi.org/10.1145/2746539.2746596
Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 86:1-86:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86
Mika Göös and Thomas Watson. Communication complexity of set-disjointness for all probabilities. Theory of Computing, 12(1):1-23, 2016. URL: http://dx.doi.org/10.4086/toc.2016.v012a009.
http://dx.doi.org/10.4086/toc.2016.v012a009
Prahladh Harsha and Rahul Jain. A strong direct product theorem for the tribes function via the smooth-rectangle bound. In Anil Seth and Nisheeth K. Vishnoi, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2013, December 12-14, 2013, Guwahati, India, volume 24 of LIPIcs, pages 141-152. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2013.141.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2013.141
Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, Cambridge, Massachusetts, June 9-12, 2010, pages 247-258. IEEE Computer Society, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.31.
http://dx.doi.org/10.1109/CCC.2010.31
Rahul Jain, Hartmut Klauck, and Shengyu Zhang. Depth-independent lower bounds on the communication complexity of read-once boolean formulas. In My T. Thai and Sartaj Sahni, editors, Computing and Combinatorics, 16th Annual International Conference, COCOON 2010, Nha Trang, Vietnam, July 19-21, 2010. Proceedings, volume 6196 of Lecture Notes in Computer Science, pages 54-59. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14031-0_8.
http://dx.doi.org/10.1007/978-3-642-14031-0_8
T. S. Jayram, Swastik Kopparty, and Prasad Raghavendra. On the communication complexity of read-once ac^0 formulae. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15-18 July 2009, pages 329-340. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/CCC.2009.39.
http://dx.doi.org/10.1109/CCC.2009.39
T. S. Jayram, Ravi Kumar, and D. Sivakumar. Two applications of information complexity. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 673-682. ACM, 2003. URL: http://dx.doi.org/10.1145/780542.780640.
http://dx.doi.org/10.1145/780542.780640
Hartmut Klauck. Rectangle size bounds and threshold covers in communication complexity. In 18th Annual IEEE Conference on Computational Complexity (Complexity 2003), 7-10 July 2003, Aarhus, Denmark, pages 118-134. IEEE Computer Society, 2003. URL: http://dx.doi.org/10.1109/CCC.2003.1214415.
http://dx.doi.org/10.1109/CCC.2003.1214415
Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997.
Nikos Leonardos and Michael E. Saks. Lower bounds on the randomized communication complexity of read-once functions. Computational Complexity, 19(2):153-181, 2010. URL: http://dx.doi.org/10.1007/s00037-010-0292-2.
http://dx.doi.org/10.1007/s00037-010-0292-2
Alexander A. Razborov. On the distributional complexity of disjointness. Theor. Comput. Sci., 106(2):385-390, 1992. URL: http://dx.doi.org/10.1016/0304-3975(92)90260-M.
http://dx.doi.org/10.1016/0304-3975(92)90260-M
Alexander A. Sherstov. The communication complexity of gap hamming distance. Theory of Computing, 8(1):197-208, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a008.
http://dx.doi.org/10.4086/toc.2012.v008a008
Thomas Vidick. A concentration inequality for the overlap of a vector on a large set, with application to the communication complexity of the Gap-Hamming-Distance problem. Chicago Journal of Theoretical Computer Science, 2012(1):1-12, 2012. URL: http://dx.doi.org/10.4086/cjtcs.2012.001.
http://dx.doi.org/10.4086/cjtcs.2012.001
Thomas Watson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity
Testing whether a set f of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). Previously, the best complexity known was NP^{#P} (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we put the problem in AM cap coAM. In particular, dependence testing is unlikely to be NP-hard and joins the league of problems of "intermediate" complexity, eg. graph isomorphism & integer factoring. Our proof method is algebro-geometric- estimating the size of the image/preimage of the polynomial map f over the finite field. A gap in this size is utilized in the AM protocols.
Next, we study the open question of testing whether every annihilator of f has zero constant term (Kayal, CCC'09). We give a geometric characterization using Zariski closure of the image of f; introducing a new problem called approximate polynomials satisfiability (APS). We show that APS is NP-hard and, using projective algebraic-geometry ideas, we put APS in PSPACE (prior best was EXPSPACE via Gröbner basis computation). As an unexpected application of this to approximative complexity theory we get- over any field, hitting-sets for overline{VP} can be verified in PSPACE. This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly mitigating the GCT Chasm (exponentially in terms of space complexity).
algebraic dependence
Jacobian
Arthur-Merlin
approximate polynomial
satisfiability
hitting-set
border VP
finite field
PSPACE
EXPSPACE
GCT Chasm
Theory of computation~Algebraic complexity theory
Theory of computation~Complexity classes
Mathematics of computing~Computations on polynomials
Mathematics of computing~Computations in finite fields
10:1-10:21
Regular Paper
Zeyu
Guo
Zeyu Guo
Department of Computer Science & Engineering, Indian Institute of Technology Kanpur
Z.G. is funded by DST and Research I Foundation of CSE, IITK.
Nitin
Saxena
Nitin Saxena
Department of Computer Science & Engineering, Indian Institute of Technology Kanpur
N.S. thanks the funding support from DST (DST/SJF/MSA-01/2013-14).
Amit
Sinhababu
Amit Sinhababu
Department of Computer Science & Engineering, Indian Institute of Technology Kanpur
A.S. thanks the travel fund support from Indian Association for Research in Computing Science and ACM India.
10.4230/LIPIcs.CCC.2018.10
L. M. Adleman and H. W. Lenstra. Finding irreducible polynomials over finite fields. In STOC, pages 350-355, 1986.
M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits. In Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), pages 599-614, 2012. (In SICOMP special issue).
Manindra Agrawal, Sumanta Ghosh, and Nitin Saxena. Bootstrapping variables in algebraic circuits, 2017. (To appear in 50th ACM Symposium on Theory of Computing (STOC), 2018). URL: https://www.cse.iitk.ac.in/users/nitin/research.html.
https://www.cse.iitk.ac.in/users/nitin/research.html
S. Arora and B. Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009.
László Babai. Trading group theory for randomness. In Proceedings of the seventeenth annual ACM symposium on Theory of computing, pages 421-429. ACM, 1985.
M. Beecken, J. Mittmann, and N. Saxena. Algebraic Independence and Blackbox Identity Testing. Inf. Comput., 222:2-19, 2013. (Conference version in ICALP 2011).
Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. On algebraic branching programs of small width. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 20:1-20:31, 2017.
Peter Bürgisser. The complexity of factors of multivariate polynomials. Foundations of Computational Mathematics, 4(4):369-396, 2004. (Preliminary version in FOCS 2001).
Peter Bürgisser, Michael Clausen, and Amin Shokrollahi. Algebraic complexity theory, volume 315. Springer Science &Business Media, 2013.
Peter Bürgisser, Ankit Garg, Rafael Mendes de Oliveira, Michael Walter, and Avi Wigderson. Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.24.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.24
Laszlo Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618-623, 1976. (Conference version in FOCS 1975).
Harm Derksen and Gregor Kemper. Computational invariant theory. Springer, 2015.
Z. Dvir, A. Gabizon, and A. Wigderson. Extractors and rank extractors for polynomial sources. Comput. Complex., 18(1):1-58, 2009. (Conference version in FOCS 2007).
Zeev Dvir. Extractors for varieties. In Proceedings of the 24th IEEE Conference on Computational Complexity (CCC), pages 102-113, 2009.
Richard Ehrenborg and Gian-Carlo Rota. Apolarity and canonical forms for homogeneous polynomials. European Journal of Combinatorics, 14(3):157-181, 1993.
Michael A. Forbes and Amir Shpilka. A PSPACE construction of a hitting set for the closure of small algebraic circuits. Electronic Colloquium on Computational Complexity (ECCC), 24:163, 2017. (To appear in 50th ACM Symposium on Theory of Computing (STOC), 2018).
Joe Harris. Algebraic Geometry: A First Course. Springer, 1992.
Robin Hartshorne. Algebraic geometry, volume 52. Springer Science &Business Media, 2013.
Joos Heintz and Claus-Peter Schnorr. Testing polynomials which are easy to compute. In Proceedings of the twelfth annual ACM symposium on Theory of computing, pages 262-272. ACM, 1980.
Aubrey W Ingleton. Representation of matroids. Combinatorial mathematics and its applications, 23, 1971.
C. G. J. Jacobi. De determinantibus functionalibus. J. Reine Angew. Math., 22(4):319-359, 1841.
N. Kayal. The Complexity of the Annihilating Polynomial. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC), pages 184-193, 2009.
Neeraj Kayal and Nitin Saxena. Complexity of ring morphism problems. computational complexity, 15(4):342-390, 2006.
Pascal Koiran. Hilbert’s Nullstellensatz is in the polynomial hierarchy. Journal of complexity, 12(4):273-286, 1996.
János Kollár. Sharp effective Nullstellensatz. Journal of the American Mathematical Society, 1(4):963-975, 1988.
Mrinal Kumar and Shubhangi Saraf. Arithmetic circuits with locally low algebraic rank. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 34:1-34:27, 2016.
Joseph M Landsberg. Tensors: geometry and applications, volume 128. American Mathematical Society Providence, RI, 2012.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th international symposium on symbolic and algebraic computation, pages 296-303. ACM, 2014.
Thomas Lehmkuhl and Thomas Lickteig. On the order of approximation in approximative triadic decompositions of tensors. Theoretical Computer Science, 66(1):1-14, 1989.
Ernst W Mayr and Albert R Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in mathematics, 46(3):305-329, 1982.
Johannes Mittmann, Nitin Saxena, and Peter Scheiblechner. Algebraic independence in positive characteristic: A p-adic calculus. Transactions of the American Mathematical Society, 366(7):3425-3450, 2014.
Ketan Mulmuley. Geometric complexity theory V: Efficient algorithms for Noether normalization. Journal of the American Mathematical Society, 30(1):225-309, 2017.
Ketan D. Mulmuley. Geometric complexity theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noether’s normalization lemma. In FOCS, pages 629-638, 2012.
Anurag Pandey, Nitin Saxena, and Amit Sinhababu. Algebraic independence over positive characteristic: New criterion and applications to locally low algebraic rank circuits. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 74:1-74:15, 2016. (In print, Computational Complexity, 2018).
O. Perron. Algebra I (Die Grundlagen). W. de Gruyter, Berlin, 1927.
Arkadiusz Płoski. Algebraic dependence of polynomials after o. perron and some applications. Computational Commutative and Non-Commutative Algebraic Geometry, pages 167-173, 2005.
Ran Raz. Elusive functions and lower bounds for arithmetic circuits. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 711-720. ACM, 2008.
Nitin Saxena. Progress on polynomial identity testing. Bulletin of the EATCS, 99:49-79, 2009.
Nitin Saxena. Progress on polynomial identity testing - II. Electronic Colloquium on Computational Complexity (ECCC), 20:186, 2013. URL: http://eccc.hpi-web.de/report/2013/186.
http://eccc.hpi-web.de/report/2013/186
Marcus Schaefer and Daniel Štefankovič. The complexity of tensor rank. Theory of Computing Systems, Aug 2017. URL: http://dx.doi.org/10.1007/s00224-017-9800-y.
http://dx.doi.org/10.1007/s00224-017-9800-y
Joachim Schmid. On the affine Bezout inequality. manuscripta mathematica, 88(1):225-232, 1995.
J.T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980.
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010.
Zeyu Guo, Nitin Saxena, and Amit Sinhababu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits
We prove a lower bound of Omega(n^2/log^2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x_1, ..., x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([Ran Raz et al., 2008]), who proved a lower bound of Omega(n^{4/3}/log^2 n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.
Algebraic Complexity
Multilinear Circuits
Circuit Lower Bounds
Theory of computation~Algebraic complexity theory
11:1-11:16
Regular Paper
https://arxiv.org/abs/1708.02037
Noga
Alon
Noga Alon
Sackler School of Mathematics and Blavatnik School of Computer Science , Tel Aviv, 6997801, Israel, and , Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Research supported in part by an ISF grant and by a GIF grant.
Mrinal
Kumar
Mrinal Kumar
Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Part of this work was done while visiting Tel Aviv University.
Ben Lee
Volk
Ben Lee Volk
Blavatnik School of Computer Science, Tel Aviv University , Tel Aviv, 6997801, Israel
The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16).
10.4230/LIPIcs.CCC.2018.11
Noga Alon, E. E. Bergmann, Don Coppersmith, and Andrew M. Odlyzko. Balancing sets of vectors. IEEE Trans. Information Theory, 34(1):128-130, 1988. URL: http://dx.doi.org/10.1109/18.2610.
http://dx.doi.org/10.1109/18.2610
Noga Alon, Mrinal Kumar, and Ben Lee Volk. An almost quadratic lower bound for syntactically multilinear arithmetic circuits. Electronic Colloquium on Computational Complexity (ECCC), 24:124, 2017. URL: https://eccc.weizmann.ac.il/report/2017/124.
https://eccc.weizmann.ac.il/report/2017/124
Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley Publishing, 4th edition, 2016. URL: http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119061954.html.
http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119061954.html
Richard P. Anstee, Lajos Rónyai, and Attila Sali. Shattering news. Graphs and Combinatorics, 18(1):59-73, 2002. URL: http://dx.doi.org/10.1007/s003730200003.
http://dx.doi.org/10.1007/s003730200003
Walter Baur and Volker Strassen. The complexity of partial derivatives. Theor. Comput. Sci., 22:317-330, 1983. URL: http://dx.doi.org/10.1016/0304-3975(83)90110-X.
http://dx.doi.org/10.1016/0304-3975(83)90110-X
Stuart J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett., 18(3):147-150, 1984. URL: http://dx.doi.org/10.1016/0020-0190(84)90018-8.
http://dx.doi.org/10.1016/0020-0190(84)90018-8
Xi Chen, Neeraj Kayal, and Avi Wigderson. Partial derivatives in arithmetic complexity and beyond. Foundations and Trends in Theoretical Computer Science, 6(1-2):1-138, 2011. URL: http://dx.doi.org/10.1561/0400000043.
http://dx.doi.org/10.1561/0400000043
L. Csanky. Fast parallel matrix inversion algorithms. SIAM J. Comput., 5(4):618-623, 1976. URL: http://dx.doi.org/10.1137/0205040.
http://dx.doi.org/10.1137/0205040
H. Enomoto, Peter Frankl, N. Ito, and K. Nomura. Codes with given distances. Graphs and Combinatorics, 3(1):25-38, 1987. URL: http://dx.doi.org/10.1007/BF01788526.
http://dx.doi.org/10.1007/BF01788526
Jeffrey B. Farr and Shuhong Gao. Computing gröbner bases for vanishing ideals of finite sets of points. In Marc P. C. Fossorier, Hideki Imai, Shu Lin, and Alain Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 16th International Symposium, AAECC-16, Las Vegas, NV, USA, February 20-24, 2006, Proceedings, volume 3857 of Lecture Notes in Computer Science, pages 118-127. Springer, 2006. URL: http://dx.doi.org/10.1007/11617983_11.
http://dx.doi.org/10.1007/11617983_11
Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth 4 formulas computing iterated matrix multiplication. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 128-135. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591824.
http://dx.doi.org/10.1145/2591796.2591824
Peter Frankl and Vojtěch Rödl. Forbidden intersections. Trans. Amer. Math. Soc., 300(1):259-286, 1987. URL: http://dx.doi.org/10.2307/2000598.
http://dx.doi.org/10.2307/2000598
Dima Grigoriev and Marek Karpinski. An exponential lower bound for depth 3 arithmetic circuits. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 577-582. ACM, 1998. URL: http://dx.doi.org/10.1145/276698.276872.
http://dx.doi.org/10.1145/276698.276872
Dima Grigoriev and Alexander A. Razborov. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Eng. Commun. Comput., 10(6):465-487, 2000. URL: http://dx.doi.org/10.1007/s002009900021.
http://dx.doi.org/10.1007/s002009900021
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. J. ACM, 61(6):33:1-33:16, 2014. URL: http://dx.doi.org/10.1145/2629541.
http://dx.doi.org/10.1145/2629541
Gábor Hegedűs. Balancing sets of vectors. Studia Sci. Math. Hungar., 47(3):333-349, 2010. URL: http://dx.doi.org/10.1556/SScMath.2009.1134.
http://dx.doi.org/10.1556/SScMath.2009.1134
Gábor Hegedűs and Lajos Rónyai. Gröbner bases for complete uniform families. J. Algebraic Combin., 17(2):171-180, 2003. URL: http://dx.doi.org/10.1023/A:1022934815185.
http://dx.doi.org/10.1023/A:1022934815185
Maurice J. Jansen. Lower bounds for syntactically multilinear algebraic branching programs. In Edward Ochmanski and Jerzy Tyszkiewicz, editors, Mathematical Foundations of Computer Science 2008, 33rd International Symposium, MFCS 2008, Torun, Poland, August 25-29, 2008, Proceedings, volume 5162 of Lecture Notes in Computer Science, pages 407-418. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-85238-4_33.
http://dx.doi.org/10.1007/978-3-540-85238-4_33
K. Kalorkoti. A lower bound for the formula size of rational functions. SIAM J. Comput., 14(3):678-687, 1985. URL: http://dx.doi.org/10.1137/0214050.
http://dx.doi.org/10.1137/0214050
Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An exponential lower bound for homogeneous depth four arithmetic formulas. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 61-70. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.15.
http://dx.doi.org/10.1109/FOCS.2014.15
Donald E. Knuth. Efficient balanced codes. IEEE Trans. Information Theory, 32(1):51-53, 1986. URL: http://dx.doi.org/10.1109/TIT.1986.1057136.
http://dx.doi.org/10.1109/TIT.1986.1057136
Mrinal Kumar. A quadratic lower bound for homogeneous algebraic branching programs. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 19:1-19:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.19.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.19
Mrinal Kumar and Ramprasad Saptharishi. An exponential lower bound for homogeneous depth-5 circuits over finite fields. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 31:1-31:30. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.31.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.31
Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 364-373. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
http://dx.doi.org/10.1109/FOCS.2014.46
Meena Mahajan and V. Vinay. Determinant: Combinatorics, algorithms, and complexity. Chicago J. Theor. Comput. Sci., 1997, 1997. Preliminary version in the \nth\intcalcSub19971989 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1997). URL: http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html.
http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html
Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 410-418. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103462.
http://dx.doi.org/10.1145/103418.103462
Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL: http://dx.doi.org/10.1007/BF01294256.
http://dx.doi.org/10.1007/BF01294256
Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(6):121-135, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a006.
http://dx.doi.org/10.4086/toc.2006.v002a006
Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2):8:1-8:17, 2009. URL: http://dx.doi.org/10.1145/1502793.1502797.
http://dx.doi.org/10.1145/1502793.1502797
Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(1):135-177, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a007.
http://dx.doi.org/10.4086/toc.2010.v006a007
Ran Raz, Amir Shpilka, and Amir Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM J. Comput., 38(4):1624-1647, 2008. URL: http://dx.doi.org/10.1137/070707932.
http://dx.doi.org/10.1137/070707932
Ran Raz and Amir Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Computational Complexity, 17(4):515-535, 2008. URL: http://dx.doi.org/10.1007/s00037-008-0254-0.
http://dx.doi.org/10.1007/s00037-008-0254-0
Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. URL: http://dx.doi.org/10.1007/s00037-009-0270-8.
http://dx.doi.org/10.1007/s00037-009-0270-8
Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, https://github.com/dasarpmar/lowerbounds-survey/, 2016. URL: https://github.com/dasarpmar/lowerbounds-survey/releases/.
https://github.com/dasarpmar/lowerbounds-survey/releases/
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
Matthew Skala. Hypergeometric tail inequalities: ending the insanity. arXiv preprint arXiv:1311.5939, 2013. URL: https://arxiv.org/abs/1311.5939.
https://arxiv.org/abs/1311.5939
Volker Strassen. Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten. Numerische Mathematik, 20(3):238-251, 1973. URL: http://dx.doi.org/10.1007/BF01436566.
http://dx.doi.org/10.1007/BF01436566
Noga Alon, Mrinal Kumar, and Ben Lee Volk
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hardness Amplification for Non-Commutative Arithmetic Circuits
We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.
This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.
arithmetic circuits
hardness amplification
circuit lower bounds
non-commutative computation
Theory of computation~Algebraic complexity theory
12:1-12:16
Regular Paper
Marco L.
Carmosino
Marco L. Carmosino
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
Russell
Impagliazzo
Russell Impagliazzo
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
Shachar
Lovett
Shachar Lovett
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by NSF CAREER award 1350481 and CCF award 1614023
Ivan
Mihajlin
Ivan Mihajlin
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
10.4230/LIPIcs.CCC.2018.12
Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75. IEEE Computer Society, 2008.
Vikraman Arvind, Pushkar S. Joglekar, Partha Mukhopadhyay, and S. Raja. Randomized polynomial time identity testing for noncommutative circuits. 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 831-841. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055442.
http://dx.doi.org/10.1145/3055399.3055442
Vikraman Arvind, Pushkar S. Joglekar, and S. Raja. Noncommutative valiant’s classes: Structure and complete problems. TOCT, 9(1):3:1-3:29, 2016. URL: http://dx.doi.org/10.1145/2956230.
http://dx.doi.org/10.1145/2956230
Vikraman Arvind and Srikanth Srinivasan. On the hardness of the noncommutative determinant. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 677-686. ACM, 2010.
Walter Baur and Volker Strassen. The complexity of partial derivatives. Theoretical computer science, 22(3):317-330, 1983.
Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. Electronic Colloquium on Computational Complexity (ECCC), 24:27, 2017.
Michael A. Forbes, Amir Shpilka, and Ben Lee Volk. Succinct hitting sets and barriers to proving algebraic circuits lower bounds. 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 653-664. ACM, 2017.
Joshua A. Grochow, Mrinal Kumar, Michael E. Saks, and Shubhangi Saraf. Towards an algebraic natural proofs barrier via polynomial identity testing. Electronic Colloquium on Computational Complexity (ECCC), 24:9, 2017.
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM J. Comput., 45(3):1064-1079, 2016.
Pavel Hrubes, Avi Wigderson, and Amir Yehudayoff. Relationless completeness and separations. Electronic Colloquium on Computational Complexity (ECCC), 17:40, 2010. URL: http://eccc.hpi-web.de/report/2010/040.
http://eccc.hpi-web.de/report/2010/040
Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011.
Pascal Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci., 448:56-65, 2012.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th international symposium on symbolic and algebraic computation, pages 296-303. ACM, 2014.
Noam Nisan. Lower bounds for non-commutative computation. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 410-418. ACM, 1991.
Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997.
Amir Shpilka, Amir Yehudayoff, et al. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trendsregistered in Theoretical Computer Science, 5(3-4):207-388, 2010.
Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL: http://dx.doi.org/10.1145/800135.804419.
http://dx.doi.org/10.1145/800135.804419
Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hardness vs Randomness for Bounded Depth Arithmetic Circuits
In this paper, we study the question of hardness-randomness tradeoffs for bounded depth arithmetic circuits. We show that if there is a family of explicit polynomials {f_n}, where f_n is of degree O(log^2n/log^2 log n) in n variables such that f_n cannot be computed by a depth Delta arithmetic circuits of size poly(n), then there is a deterministic sub-exponential time algorithm for polynomial identity testing of arithmetic circuits of depth Delta-5.
This is incomparable to a beautiful result of Dvir et al.[SICOMP, 2009], where they showed that super-polynomial lower bounds for depth Delta circuits for any explicit family of polynomials (of potentially high degree) implies sub-exponential time deterministic PIT for depth Delta-5 circuits of bounded individual degree. Thus, we remove the "bounded individual degree" condition in the work of Dvir et al. at the cost of strengthening the hardness assumption to hold for polynomials of low degree.
The key technical ingredient of our proof is the following property of roots of polynomials computable by a bounded depth arithmetic circuit : if f(x_1, x_2, ..., x_n) and P(x_1, x_2, ..., x_n, y) are polynomials of degree d and r respectively, such that P can be computed by a circuit of size s and depth Delta and P(x_1, x_2, ..., x_n, f) equiv 0, then, f can be computed by a circuit of size poly(n, s, r, d^{O(sqrt{d})}) and depth Delta + 3. In comparison, Dvir et al. showed that f can be computed by a circuit of depth Delta + 3 and size poly(n, s, r, d^{t}), where t is the degree of P in y. Thus, the size upper bound in the work of Dvir et al. is non-trivial when t is small but d could be large, whereas our size upper bound is non-trivial when d is small, but t could be large.
Algebraic Complexity
Polynomial Factorization Circuit Lower Bounds
Polynomial Identity Testing
Theory of computation~Algebraic complexity theory
13:1-13:17
Regular Paper
https://eccc.weizmann.ac.il/report/2018/052/
Chi-Ning
Chou
Chi-Ning Chou
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Mrinal
Kumar
Mrinal Kumar
Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Noam
Solomon
Noam Solomon
Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
10.4230/LIPIcs.CCC.2018.13
Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.32.
http://dx.doi.org/10.1109/FOCS.2008.32
Walter Baur and Volker Strassen. The complexity of partial derivatives. Theor. Comput. Sci., 22:317-330, 1983. URL: http://dx.doi.org/10.1016/0304-3975(83)90110-X.
http://dx.doi.org/10.1016/0304-3975(83)90110-X
Peter Bürgisser. The complexity of factors of multivariate polynomials. Foundations of Computational Mathematics, 4(4):369-396, 2004. URL: http://dx.doi.org/10.1007/s10208-002-0059-5.
http://dx.doi.org/10.1007/s10208-002-0059-5
Pranjal Dutta, Nitin Saxena, and Amit Sinhababu. Discovering the roots: Uniform closure results for algebraic classes under factoring. CoRR, abs/1710.03214, 2017. URL: http://arxiv.org/abs/1710.03214.
http://arxiv.org/abs/1710.03214
Zeev Dvir, Amir Shpilka, and Amir Yehudayoff. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. Comput., 39(4):1279-1293, 2009. URL: http://dx.doi.org/10.1137/080735850.
http://dx.doi.org/10.1137/080735850
Michael A. Forbes. Deterministic divisibility testing via shifted partial derivatives. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 451-465. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.35.
http://dx.doi.org/10.1109/FOCS.2015.35
Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth 4 formulas computing iterated matrix multiplication. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 128-135. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591824.
http://dx.doi.org/10.1145/2591796.2591824
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth three. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 578-587. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.68.
http://dx.doi.org/10.1109/FOCS.2013.68
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. J. ACM, 61(6):33:1-33:16, 2014. URL: http://dx.doi.org/10.1145/2629541.
http://dx.doi.org/10.1145/2629541
Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. URL: http://dx.doi.org/10.1007/s00037-004-0182-6.
http://dx.doi.org/10.1007/s00037-004-0182-6
K. Kalorkoti. A lower bound for the formula size of rational functions. SIAM J. Comput., 14(3):678-687, 1985. URL: http://dx.doi.org/10.1137/0214050.
http://dx.doi.org/10.1137/0214050
Erich Kaltofen. Factorization of Polynomials Given by Straight-Line Programs. In Randomness and Computation, pages 375-412. JAI Press, 1989.
Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 146-153. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591847.
http://dx.doi.org/10.1145/2591796.2591847
Pascal Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci., 448:56-65, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.03.041.
http://dx.doi.org/10.1016/j.tcs.2012.03.041
Mrinal Kumar. A quadratic lower bound for homogeneous algebraic branching programs. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 19:1-19:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.19.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.19
Mrinal Kumar and Ramprasad Saptharishi. An exponential lower bound for homogeneous depth-5 circuits over finite fields. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 31:1-31:30. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.31.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.31
Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 364-373. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
http://dx.doi.org/10.1109/FOCS.2014.46
Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80043-1.
http://dx.doi.org/10.1016/S0022-0000(05)80043-1
Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL: http://dx.doi.org/10.1007/BF01294256.
http://dx.doi.org/10.1007/BF01294256
Rafael Oliveira. Factors of low individual degree polynomials. Computational Complexity, 25(2):507-561, 2016. URL: http://dx.doi.org/10.1007/s00037-016-0130-2.
http://dx.doi.org/10.1007/s00037-016-0130-2
Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(6):121-135, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a006.
http://dx.doi.org/10.4086/toc.2006.v002a006
Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(1):135-177, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a007.
http://dx.doi.org/10.4086/toc.2010.v006a007
Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. URL: http://dx.doi.org/10.1145/2535928.
http://dx.doi.org/10.1145/2535928
Ran Raz, Amir Shpilka, and Amir Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM J. Comput., 38(4):1624-1647, 2008. URL: http://dx.doi.org/10.1137/070707932.
http://dx.doi.org/10.1137/070707932
Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. URL: http://dx.doi.org/10.1007/s00037-009-0270-8.
http://dx.doi.org/10.1007/s00037-009-0270-8
Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, https://github.com/dasarpmar/lowerbounds-survey/, 2016. URL: https://github.com/dasarpmar/lowerbounds-survey/releases/.
https://github.com/dasarpmar/lowerbounds-survey/releases/
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Inf. Comput., 240:2-11, 2015. URL: http://dx.doi.org/10.1016/j.ic.2014.09.004.
http://dx.doi.org/10.1016/j.ic.2014.09.004
Chi-Ning Chou, Mrinal Kumar, and Noam Solomon
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product
In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a in A and b in B maximizing inner product a * b. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact l_2-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three main results regarding this problem.
- Characterization of Multiplicative Approximation. First, we study the best multiplicative approximation ratio for Boolean Max-IP in sub-quadratic time. We show that, for Max-IP with two sets of n vectors from {0,1}^{d}, there is an n^{2 - Omega(1)} time (d/log n)^{Omega(1)}-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/log n)^{o(1)}-approximating algorithm would refute SETH. Similar characterization is also achieved for additive approximation for Max-IP.
- 2^{O(log^* n)}-dimensional Hardness for Exact Max-IP Over The Integers. Second, we revisit the hardness of solving Max-IP exactly for vectors with integer entries. We show that, under SETH, for Max-IP with sets of n vectors from Z^{d} for some d = 2^{O(log^* n)}, every exact algorithm requires n^{2 - o(1)} time. With the reduction from [Williams, SODA 2018], it follows that l_2-Furthest Pair and Bichromatic l_2-Closest Pair in 2^{O(log^* n)} dimensions require n^{2 - o(1)} time.
- Connection with NP * UPP Communication Protocols. Last, We establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP * UPP communication protocols for Set-Disjointness, parallel to the connection between conditional lower bounds for approximating Max-IP and MA communication protocols for Set-Disjointness.
The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018], and our algorithms utilize the polynomial method and simple random sampling. Our second result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for n vectors from {0,1}^{d} to n vectors from Z^{l} where l = 2^{O(log^* d)}, dramatically improving the previous reduction in [Williams, SODA 2018]. The key technical ingredient is a recursive application of Chinese Remainder Theorem.
As a side product, we obtain an MA communication protocol for Set-Disjointness with complexity O (sqrt{n log n log log n}), slightly improving the O (sqrt{n} log n) bound [Aaronson and Wigderson, TOCT 2009], and approaching the Omega(sqrt{n}) lower bound [Klauck, CCC 2003].
Moreover, we show that (under SETH) one can apply the O(sqrt{n}) BQP communication protocol for Set-Disjointness to prove near-optimal hardness for approximation to Max-IP with vectors in {-1,1}^d. This answers a question from [Abboud et al., FOCS 2017] in the affirmative.
Maximum Inner Product
SETH
Hardness of Approximation in P
Fined-Grained Complexity
Hopcroft's Problem
Chinese Remainder Theorem
Theory of computation~Problems, reductions and completeness
14:1-14:45
Regular Paper
Lijie
Chen
Lijie Chen
Massachusetts Institute of Technology, USA
Supported by an Akamai Fellowship
10.4230/LIPIcs.CCC.2018.14
Scott Aaronson and Avi Wigderson. Algebrization: A new barrier in complexity theory. TOCT, 1(1):2:1-2:54, 2009. URL: http://dx.doi.org/10.1145/1490270.1490272.
http://dx.doi.org/10.1145/1490270.1490272
Amir Abboud and Arturs Backurs. Towards hardness of approximation for polynomial time problems. In LIPIcs-Leibniz International Proceedings in Informatics, volume 67. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
Amir Abboud and Søren Dahlgaard. Popular conjectures as a barrier for dynamic planar graph algorithms. In Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, pages 477-486, 2016.
Amir Abboud and Aviad Rubinstein. Fast and deterministic constant factor approximation algorithms for lcs imply new circuit lower bounds. In LIPIcs-Leibniz International Proceedings in Informatics, volume 94. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
Amir Abboud, Aviad Rubinstein, and R. Ryan Williams. Distributed PCP theorems for hardness of approximation in P. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 25-36. IEEE Computer Society, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.12.
http://dx.doi.org/10.1109/FOCS.2017.12
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. of the 55th FOCS, pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Proc. of the 41st ICALP, pages 39-51, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 41-50. ACM, 2015.
Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 218-230. Society for Industrial and Applied Mathematics, 2015.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proceedings of the IEEE 55th Annual Symposium on Foundations of Computer Science, pages 434-443, 2014.
Pankaj K Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean minimum spanning trees and bichromatic closest pairs. Discrete &Computational Geometry, 6(3):407-422, 1991.
Thomas Dybdahl Ahle, Rasmus Pagh, Ilya Razenshteyn, and Francesco Silvestri. On the complexity of inner product similarity join. In Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 151-164. ACM, 2016.
Josh Alman, Timothy M Chan, and Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 467-476. IEEE, 2016.
Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In Proc. of the 56th FOCS, pages 136-150. IEEE, 2015.
Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In Proc. of the 47th FOCS, pages 459-468. IEEE, 2006.
Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, and Ludwig Schmidt. Practical and optimal lsh for angular distance. In Advances in Neural Information Processing Systems, pages 1225-1233, 2015.
Alexandr Andoni, Piotr Indyk, Huy L Nguyen, and Ilya Razenshteyn. Beyond locality-sensitive hashing. In Proc. of the 25th SODA, pages 1018-1028. SIAM, 2014.
Alexandr Andoni and Ilya Razenshteyn. Optimal data-dependent hashing for approximate near neighbors. In Proc. of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 793-801. ACM, 2015.
Tom M. Apostol. Introduction to analytic number theory. Springer Science &Business Media, 2013.
Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264.
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264
Arturs Backurs and Piotr Indyk. Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false). In Proc. of the 47th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 51-58, 2015.
Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? In Proc. of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 457-466, 2016.
Jon Louis Bentley and Michael Ian Shamos. Divide-and-conquer in multidimensional space. In Proceedings of the eighth annual ACM symposium on Theory of computing, pages 220-230. ACM, 1976.
Karl Bringman and Marvin Künnemann. Multivariate fine-grained complexity of longest common subsequence. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1216-1235. SIAM, 2018.
Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 661-670, 2014.
Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A dichotomy for regular expression membership testing. arXiv preprint arXiv:1611.00918, 2016.
Harry Buhrman, Richard Cleve, Ronald De Wolf, and Christof Zalka. Bounds for small-error and zero-error quantum algorithms. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 358-368. IEEE, 1999.
Harry Buhrman, Richard Cleve, and Avi Wigderson. Quantum vs. classical communication and computation. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 63-68. ACM, 1998.
Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In IWPEC, volume 5917, pages 75-85. Springer, 2009.
Timothy M Chan. A (slightly) faster algorithm for klee’s measure problem. In Proceedings of the twenty-fourth annual symposium on Computational geometry, pages 94-100. ACM, 2008.
Tobias Christiani. A framework for similarity search with space-time tradeoffs using locality-sensitive filtering. In Proc. of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 31-46. SIAM, 2017.
Tobias Christiani and Rasmus Pagh. Set similarity search beyond minhash. arXiv preprint arXiv:1612.07710, 2016.
Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM Journal on Computing, 11(3):467-471, 1982.
Svyatoslav Covanov and Emmanuel Thomé. Fast integer multiplication using generalized fermat primes. arXiv preprint arXiv:1502.02800, 2015.
Roee David, CS Karthik, and Bundit Laekhanukit. On the complexity of closest pair via polar-pair of point-sets. CoRR, abs/1608.03245, 2016.
Ronald de Wolf. A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions. arXiv preprint arXiv:0802.1816, 2008.
Martin Dietzfelbinger, Torben Hagerup, Jyrki Katajainen, and Martti Penttonen. A reliable randomized algorithm for the closest-pair problem. Journal of Algorithms, 25(1):19-51, 1997.
Martin Fürer. Faster integer multiplication. SIAM Journal on Computing, 39(3):979-1005, 2009.
Francois Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using powers of the coppersmith-winograd tensor. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1029-1046. SIAM, 2018.
Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and R. Ryan Williams. Completeness for first-order properties on sparse structures with algorithmic applications. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2162-2181, 2017.
Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. Conditional lower bounds for space/time tradeoffs. In Faith Ellen, Antonina Kolokolova, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures, pages 421-436, Cham, 2017. Springer International Publishing.
Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212-219. ACM, 1996.
David Harvey, Joris Van Der Hoeven, and Grégoire Lecerf. Even faster integer multiplication. Journal of Complexity, 36:1-30, 2016.
Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 21-30, 2015.
Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional hardness for sensitivity problems. arXiv preprint arXiv:1703.01638, 2017.
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
http://dx.doi.org/10.1006/jcss.2000.1727
Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proc. of the thirtieth annual ACM symposium on Theory of computing, pages 604-613. ACM, 1998.
Stasys Jukna. Boolean function complexity: advances and frontiers, volume 27. Springer Science &Business Media, 2012.
Matti Karppa, Petteri Kaski, and Jukka Kohonen. A faster subquadratic algorithm for finding outlier correlations. In Proc. of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1288-1305. Society for Industrial and Applied Mathematics, 2016.
C.S. Karthik, Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. arXiv preprint arXiv:1711.11029, 2017.
Samir Khuller and Yossi Matias. A simple randomized sieve algorithm for the closest-pair problem. Information and Computation, 118(1):34-37, 1995.
Hartmut Klauck. Rectangle size bounds and threshold covers in communication complexity. In Computational Complexity, 2003. Proceedings. 18th IEEE Annual Conference on, pages 118-134. IEEE, 2003.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1272-1287, 2016.
Robert Krauthgamer and Ohad Trabelsi. Conditional lower bounds for all-pairs max-flow. arXiv preprint arXiv:1702.05805, 2017.
Jiří Matoušek. Efficient partition trees. Discrete &Computational Geometry, 8(3):315-334, 1992.
Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete &Computational Geometry, 10(2):157-182, 1993.
Behnam Neyshabur and Nathan Srebro. On symmetric and asymmetric lshs for inner product search. In Proc. of the 32nd International Conference on Machine Learning, ICML, pages 1926-1934, 2015.
Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 603-610, 2010.
Mihai Pătraşcu and Ryan Williams. On the possibility of faster sat algorithms. In Proc. of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1065-1075. SIAM, 2010.
Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. Journal of Computer and System Sciences, 33(1):106-123, 1986.
Ali Rahimi, Benjamin Recht, et al. Random features for large-scale kernel machines. In NIPS, volume 3, page 5, 2007.
Parikshit Ram and Alexander G Gray. Maximum inner-product search using cone trees. In Proc. of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 931-939. ACM, 2012.
Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proc. of the 45th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 515-524, 2013.
Aviad Rubinstein. Hardness of approximate nearest neighbor search. In STOC, page To appear, 2018.
Anshumali Shrivastava and Ping Li. Asymmetric lsh (alsh) for sublinear time maximum inner product search (mips). In Advances in Neural Information Processing Systems, pages 2321-2329, 2014.
Anshumali Shrivastava and Ping Li. Asymmetric minwise hashing for indexing binary inner products and set containment. In Proc. of the 24th International Conference on World Wide Web, pages 981-991. ACM, 2015.
Christina Teflioudi and Rainer Gemulla. Exact and approximate maximum inner product search with lemp. ACM Transactions on Database Systems (TODS), 42(1):5, 2016.
Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and the closest pair problem. Journal of the ACM (JACM), 62(2):13, 2015.
Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In To appear in the proceedings of the ICM, 2018.
R. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005.
Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 664-673. ACM, 2014.
Ryan Williams. On the difference between closest, furthest, and orthogonal pairs: Nearly-linear vs barely-subquadratic complexity. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1207-1215. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.78.
http://dx.doi.org/10.1137/1.9781611975031.78
Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 1867-1877. SIAM, 2014.
Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982.
Lijie Chen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hardness of Function Composition for Semantic Read once Branching Programs
In this work, we study time/space trade-offs for function composition. We prove asymptotically optimal lower bounds for function composition in the setting of nondeterministic read once branching programs, for the syntactic model as well as the stronger semantic model of read-once nondeterministic computation. We prove that such branching programs for solving the tree evaluation problem over an alphabet of size k requires size roughly k^{Omega(h)}, i.e space Omega(h log k). Our lower bound nearly matches the natural upper bound which follows the best strategy for black-white pebbling the underlying tree. While previous super-polynomial lower bounds have been proven for read-once nondeterministic branching programs (for both the syntactic as well as the semantic models), we give the first lower bounds for iterated function composition, and in these models our lower bounds are near optimal.
Branching Programs
Function Composition
Time-Space Tradeoffs
Semantic Read Once
Tree Evaluation Problem
Theory of computation~Complexity classes
15:1-15:22
Regular Paper
Research supported by NSERC
Jeff
Edmonds
Jeff Edmonds
York University, 4700 Keele Street, Toronto, CANADA
http://www.cs.yorku.ca/ jeff/
Venkatesh
Medabalimi
Venkatesh Medabalimi
University of Toronto, 10 King’s College Road, Toronto, CANADA
https://www.cs.toronto.edu/ venkatm
Toniann
Pitassi
Toniann Pitassi
University of Toronto, 10 King’s College Road, Toronto, CANADA, and Institute for Advanced Study, Princeon NJ
https://www.cs.toronto.edu/ toni/
10.4230/LIPIcs.CCC.2018.15
M. Ajtai. A non-linear time lower bound for boolean branching programs. In Proceedings 40th FOCS, pages 60-70, 1999.
P. Beame, T.S. Jayram, and M. Saks. Time-space tradeoffs for branching programs. J. Comput. Syst. Sci, 63(4):542-572, 2001.
P. Beame, M. Saks, X. Sun, and E. Vee. Time-space trade-off lower bounds for randomized computation of decision problems. Journal of the ACM, 50(2):154-195, 2003.
Paul Beame, Nathan Grosshans, Pierre McKenzie, and Luc Segoufin. Nondeterminism and an abstract formulation of nečiporuk’s lower bound method. ACM Transactions on Computation Theory, 9, 08 2016.
Paul Beame and Pierre McKenzie. A note on neciporuk’s method for nondeterministic branching programs. Manuscript, August, 2011.
Allan Borodin, A Razborov, and Roman Smolensky. On lower bounds for read-k-times branching programs. Computational Complexity, 3(1):1-18, 1993.
Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration inequalities: A nonasymptotic theory of independence. Oxford university press, 2013.
Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. J. ACM, 63(4):34:1-34:22, 2016. URL: http://dx.doi.org/10.1145/2811255.
http://dx.doi.org/10.1145/2811255
Stephen Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam. Pebbles and branching programs for tree evaluation. ACM Transactions on Computation Theory (TOCT), 3(2):4, 2012.
Stephen Cook and Ravi Sethi. Storage requirements for deterministic polynomialtime recognizable languages. Journal of Computer and System Sciences, 13(1):25-37, 1976.
Stephen A. Cook, Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi. Lower bounds for nondeterministic semantic read-once branching programs. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 36:1-36:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.36.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.36
Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. How limited interaction hinders real communication (and what it means for proof and circuit complexity). In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 295-304, 2016.
Scott Diehl and Dieter Van Melkebeek. Time-space lower bounds for the polynomial-time hierarchy on randomized machines. SIAM Journal on Computing, 36(3):563-594, 2006.
Irit Dinur and Or Meir. Toward the krw composition conjecture: Cubic formula lower bounds via communication complexity. In LIPIcs-Leibniz International Proceedings in Informatics, volume 50. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016.
Jeff Edmonds, Russell Impagliazzo, Steven Rudich, and Jiri Sgall. Communication complexity towards lower bounds on circuit depth. Computational Complexity, 10(3):210-246, 2001.
L. Fortnow. Nondeterministic polynomial time versus nondeterministic logarithmic space: Time space tradeoffs for satifiability. In Proceedings 12th Conference on Computational Complexity, pages 52-60, 1997.
L. Fortnow and D. Van Melkebeek. Time-space tradeoffs for nondeterministic computation. In Proceedings 15th Conference on Computational Complexity, pages 2-13, 2000.
Lance Fortnow, Richard Lipton, Dieter Van Melkebeek, and Anastasios Viglas. Time-space lower bounds for satisfiability. Journal of the ACM (JACM), 52(6):835-865, 2005.
Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward better formula lower bounds: an information complexity approach to the krw composition conjecture. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 213-222. ACM, 2014.
Mika Göös. Lower bounds for clique vs. independent set. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1066-1076, 2015.
S. Jukna. A nondeterministic space-time tradeoff for linear codes. Information Processing Letters, 109(5):286-289, 2009.
Stasys Jukna. A note on read-k times branching programs. Informatique théorique et applications, 29(1):75-83, 1995.
Stasys Jukna. Boolean function complexity: advances and frontiers, volume 27. Springer Science &Business Media, 2012.
Stasys P Jukna. The effect of null-chains on the complexity of contact schemes. In Fundamentals of Computation Theory, pages 246-256. Springer, 1989.
Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3/4):191-204, 1995. URL: http://dx.doi.org/10.1007/BF01206317.
http://dx.doi.org/10.1007/BF01206317
Pravesh K. Kothari, Raghu Meka, and Prasad Raghavendra. Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of csps. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 590-603, 2017.
Matthias Krause, Christoph Meinel, and Stephan Waack. Separating the eraser turing machine classes le, nle, co-nle and pe. In Mathematical Foundations of Computer Science 1988, pages 405-413. Springer, 1988.
James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 567-576. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746599.
http://dx.doi.org/10.1145/2746539.2746599
R. Lipton and A. Viglas. Time-space tradeoffs for sat. In Proceedings 40th FOCS, pages 459-464, 1999.
David Liu. Pebbling arguments for tree evaluation. CoRR, abs/1311.0293, 2013. URL: http://arxiv.org/abs/1311.0293.
http://arxiv.org/abs/1311.0293
Edward I Nechiporuk. On a boolean function. Doklady Akademii Nauk SSSR, 169(4):765-+, 1966.
EA Okolnishnikova. On lower bounds for branching programs. Siberian Advances in Mathematics, 3(1):152-166, 1993.
Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, 1999. URL: http://dx.doi.org/10.1007/s004930050062.
http://dx.doi.org/10.1007/s004930050062
Johan Håstad. The shrinkage exponent of de morgan formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998.
Avishay Tal. Shrinkage of de morgan formulae by spectral techniques. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 551-560. IEEE, 2014.
Ryan Williams. Better time-space lower bounds for sat and related problems. In Computational Complexity, 2005. Proceedings. Twentieth Annual IEEE Conference on, pages 40-49. IEEE, 2005.
Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Reordering Rule Makes OBDD Proof Systems Stronger
Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(^, weakening), simulates CP^* (Cutting Planes with unary coefficients). We show that OBDD(^, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(^, weakening) system.
The reordering rule allows changing the variable order for OBDDs. We show that OBDD(^, weakening, reordering) is strictly stronger than OBDD(^, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(^) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema.
Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP^*, OBDD(^), OBDD(^, reordering), OBDD(^, weakening) and OBDD(^, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential.
Proof complexity
OBDD
Tseitin formulas
the Clique-Coloring principle
lifting theorems
Theory of computation~Computational complexity and cryptography
16:1-16:24
Regular Paper
The research was supported by the Russian Science Foundation (project 16-11-10123)
Sam
Buss
Sam Buss
University of California, San Diego, La Jolla, CA, USA
Dmitry
Itsykson
Dmitry Itsykson
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia
Alexander
Knop
Alexander Knop
University of California, San Diego, La Jolla, CA, USA, St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia
Dmitry
Sokolov
Dmitry Sokolov
KTH Royal Institute of Technology, Stockholm, Sweden , St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia
10.4230/LIPIcs.CCC.2018.16
Albert Atserias, Phokion G. Kolaitis, and Moshe Y. Vardi. Constraint propagation as a proof system. In Mark Wallace, editor, Principles and Practice of Constraint Programming - CP 2004, volume 3258 of Lecture Notes in Computer Science, pages 77-91. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-540-30201-8_9.
http://dx.doi.org/10.1007/978-3-540-30201-8{_}9
Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001. URL: http://dx.doi.org/10.1145/375827.375835.
http://dx.doi.org/10.1145/375827.375835
Randal E. Bryant. Symbolic manipulation of boolean functions using a graphical representation. In Hillel Ofek and Lawrence A. O'Neill, editors, Proceedings of the 22nd ACM/IEEE conference on Design automation, DAC 1985, Las Vegas, Nevada, USA, 1985., pages 688-694. ACM, 1985. URL: http://dx.doi.org/10.1145/317825.317964.
http://dx.doi.org/10.1145/317825.317964
Jerry R. Burch, Edmund M. Clarke, Kenneth L. McMillan, David L. Dill, and L. J. Hwang. Symbolic model checking: 10\^20 states and beyond. Inf. Comput., 98(2):142-170, 1992. URL: http://dx.doi.org/10.1016/0890-5401(92)90017-A.
http://dx.doi.org/10.1016/0890-5401(92)90017-A
Joshua Buresh-Oppenheim and Toniann Pitassi. The complexity of resolution refinements. J. Symb. Log., 72(4):1336-1352, 2007. URL: http://dx.doi.org/10.2178/jsl/1203350790.
http://dx.doi.org/10.2178/jsl/1203350790
Wei Chen and Wenhui Zhang. A direct construction of polynomial-size OBDD proof of pigeon hole problem. Inf. Process. Lett., 109(10):472-477, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2009.01.006.
http://dx.doi.org/10.1016/j.ipl.2009.01.006
Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. Electronic Colloquium on Computational Complexity (ECCC), 24:175, 2017. URL: https://eccc.weizmann.ac.il/report/2017/175.
https://eccc.weizmann.ac.il/report/2017/175
Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 847-856. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591838.
http://dx.doi.org/10.1145/2591796.2591838
Dima Grigoriev, Edward A. Hirsch, and Dmitrii V. Pasechnik. Complexity of semi-algebraic proofs. In Helmut Alt and Afonso Ferreira, editors, STACS 2002, 19th Annual Symposium on Theoretical Aspects of Computer Science, Antibes - Juan les Pins, France, March 14-16, 2002, Proceedings, volume 2285 of Lecture Notes in Computer Science, pages 419-430. Springer, 2002. URL: http://dx.doi.org/10.1007/3-540-45841-7_34.
http://dx.doi.org/10.1007/3-540-45841-7_34
Jan Friso Groote and Hans Zantema. Resolution and binary decision diagrams cannot simulate each other polynomially. Discrete Applied Mathematics, 130(2):157-171, 2003. URL: http://dx.doi.org/10.1016/S0166-218X(02)00403-1.
http://dx.doi.org/10.1016/S0166-218X(02)00403-1
Dmitry Itsykson, Alexander Knop, Andrei E. Romashchenko, and Dmitry Sokolov. On obdd-based algorithms and proof systems that dynamically change order of variables. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, volume 66 of LIPIcs, pages 43:1-43:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2017.43.
http://dx.doi.org/10.4230/LIPIcs.STACS.2017.43
Matti Järvisalo. On the relative efficiency of DPLL and obdds with axiom and join. In Jimmy Ho-Man Lee, editor, Principles and Practice of Constraint Programming - CP 2011 - 17th International Conference, CP 2011, Perugia, Italy, September 12-16, 2011. Proceedings, volume 6876 of Lecture Notes in Computer Science, pages 429-437. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-23786-7_33.
http://dx.doi.org/10.1007/978-3-642-23786-7_33
Jan Krajícek. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symb. Log., 62(2):457-486, 1997. URL: http://dx.doi.org/10.2307/2275541.
http://dx.doi.org/10.2307/2275541
Jan Krajícek. An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams. J. Symb. Log., 73(1):227-237, 2008. URL: http://dx.doi.org/10.2178/jsl/1208358751.
http://dx.doi.org/10.2178/jsl/1208358751
Kenneth L. McMillan. Symbolic model checking. Kluwer, 1993.
Christoph Meinel and Anna Slobodova. On the complexity of constructing optimal ordered binary decision diagrams. In Proceedings of Mathematical Foundations of Computer Science, volume 841, pages 515-524, 1994.
Guoqiang Pan and Moshe Y. Vardi. Search vs. symbolic techniques in satisfiability solving. In 7th International Conference on Theory and Applications of Satisfiability Testing, SAT 2004, Revised Selected Papers, volume 3542, pages 235-250, 2005. URL: http://dx.doi.org/10.1007/11527695_19.
http://dx.doi.org/10.1007/11527695{_}19
Pavel Pudlák. Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic, 62(3):981-998, 1997.
Nathan Segerlind. On the relative efficiency of resolution-like proofs and ordered binary decision diagram proofs. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC 2008, 23-26 June 2008, College Park, Maryland, USA, pages 100-111. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/CCC.2008.34.
http://dx.doi.org/10.1109/CCC.2008.34
Dmitry Sokolov. Dag-like communication and its applications. In Pascal Weil, editor, Computer Science - Theory and Applications - 12th International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8-12, 2017, Proceedings, volume 10304 of Lecture Notes in Computer Science, pages 294-307. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-58747-9_26.
http://dx.doi.org/10.1007/978-3-319-58747-9_26
Olga Tveretina, Carsten Sinz, and Hans Zantema. Ordered binary decision diagrams, pigeonhole formulas and beyond. JSAT, 7(1):35-58, 2010. URL: http://jsat.ewi.tudelft.nl/content/volume7/JSAT7_3_Tveretina.pdf.
http://jsat.ewi.tudelft.nl/content/volume7/JSAT7_3_Tveretina.pdf
Tomás E. Uribe and Mark E. Stickel. Ordered binary decision diagrams and the davis-putnam procedure. In Jean-Pierre Jouannaud, editor, Constraints in Computational Logics, First International Conference, CCL'94, Munich, Germant, September 7-9, 1994, volume 845 of Lecture Notes in Computer Science, pages 34-49. Springer, 1994. URL: http://dx.doi.org/10.1007/BFb0016843.
http://dx.doi.org/10.1007/BFb0016843
Alasdair Urquhart. Hard examples for resolution. J. ACM, 34(1):209-219, 1987. URL: http://dx.doi.org/10.1145/7531.8928.
http://dx.doi.org/10.1145/7531.8928
Sam Buss, Dmitry Itsykson, Alexander Knop, and Dmitry Sokolov
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Testing Linearity against Non-Signaling Strategies
Non-signaling strategies are collections of distributions with certain non-local correlations. They have been studied in Physics as a strict generalization of quantum strategies to understand the power and limitations of Nature's apparent non-locality. Recently, they have received attention in Theoretical Computer Science due to connections to Complexity and Cryptography.
We initiate the study of Property Testing against non-signaling strategies, focusing first on the classical problem of linearity testing (Blum, Luby, and Rubinfeld; JCSS 1993). We prove that any non-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions.
Quasi-distributions generalize the notion of probability distributions over global objects (such as functions) by allowing negative probabilities, while at the same time requiring that "local views" follow standard distributions (with non-negative probabilities). Quasi-distributions arise naturally in the study of Quantum Mechanics as a tool to describe various non-local phenomena.
Our analysis of the linearity test relies on Fourier analytic techniques applied to quasi-distributions. Along the way, we also establish general equivalences between non-signaling strategies and quasi-distributions, which we believe will provide a useful perspective on the study of Property Testing against non-signaling strategies beyond linearity testing.
property testing
linearity testing
non-signaling strategies
quasi-distributions
Theory of computation~Computational complexity and cryptography
17:1-17:37
Regular Paper
This work was supported by the UC Berkeley Center for Long-Term Cybersecurity.
https://eccc.weizmann.ac.il/report/2018/067/
Alessandro
Chiesa
Alessandro Chiesa
UC Berkeley, Berkeley (CA), USA
Peter
Manohar
Peter Manohar
UC Berkeley, Berkeley (CA), USA
Igor
Shinkar
Igor Shinkar
UC Berkeley, Berkeley (CA), USA
10.4230/LIPIcs.CCC.2018.17
William Aiello, Sandeep N. Bhatt, Rafail Ostrovsky, and Sivaramakrishnan Rajagopalan. Fast verification of any remote procedure call: Short witness-indistinguishable one-round proofs for NP. In Proceedings of the 27th International Colloquium on Automata, Languages and Programming, ICALP '00, pages 463-474, 2000.
Sabri W. Al-Safi and Anthony J. Short. Simulating all nonsignaling correlations via classical or quantum theory with negative probabilities. Physical Review Letters, 111:170403, 2013.
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, 1998. Preliminary version in FOCS '92.
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM, 45(1):70-122, 1998. Preliminary version in FOCS '92.
Sanjeev Arora and Madhu Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365-426, 2003. Preliminary version appeared in STOC '97.
László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Proceedings of the 23rd ACM Symposium on Theory of Computing, STOC '91, pages 21-32, 1991.
László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. Preliminary version appeared in FOCS '90.
Jonathan Barrett. Information processing in generalized probabilistic theories. Physical Review A, 75:032304, 2007.
Jonathan Barrett, Lucien Hardy, and Adrian Kent. No signaling and quantum key distribution. Physical Review Letters, 95:010503, 2005.
Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. Nonlocal correlations as an information-theoretic resource. Physical Review Letters, 71:022101, 2005.
Jonathan Barrett and Stefano Pironio. Popescu-Rohrlich correlations as a unit of nonlocality. Physical Review Letters, 95:140401, 2005.
Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, and Madhu Sudan. Linearity testing in characteristic two. IEEE Transactions on Information Theory, 42(6):1781-1795, 1996.
Michael Ben-Or, Don Coppersmith, Mike Luby, and Ronitt Rubinfeld. Non-abelian homomorphism testing, and distributions close to their self-convolutions. Random Structures and Algorithms, 32(1):49-70, 2008.
Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of Reed-Muller codes. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science, FOCS '10, pages 488-497, 2010.
Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993.
Gilles Brassard, Harry Buhrman, Noah Linden, André Allan Méthot, Alain Tapp, and Falk Unger. Limit on nonlocality in any world in which communication complexity is not trivial. Physical Review Letters, 96:250401, 2006.
Anne Broadbent and André Allan Méthot. On the power of non-local boxes. Theoretical Computer Science, 358(1):3-14, 2006.
Harry Buhrman, Matthias Christandl, Falk Unger, Stephanie Wehner, and Andreas Winter. Implications of superstrong non-locality for cryptography. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 462(2071):1919-1932, 2006.
Nicolas J. Cerf, Nicolas Gisin, Serge Massar, and Sandu Popescu. Simulating maximal quantum entanglement without communication. Physical Review Letters, 94:220403, 2005.
Rui Chao and Ben W. Reichardt. Test to separate quantum theory from non-signaling theories. arXiv quant-ph/1706.02008, 2017.
Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. SIAM Journal on Computing, 46:1336-1369, 2017.
Paul A. M. Dirac. The physical interpretation of quantum mechanics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 180(980):1-40, 1942.
Cynthia Dwork, Michael Langberg, Moni Naor, Kobbi Nissim, and Omer Reingold. Succinct NP proofs and spooky interactions, December 2004. Available at URL: https://www.openu.ac.il/home/mikel/papers/spooky.ps.
https://www.openu.ac.il/home/mikel/papers/spooky.ps
Uriel Feige, Shafi Goldwasser, Laszlo Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292, 1996. Preliminary version in FOCS '91.
Richard P. Feynman. Negative probability. In Basil J. Hiley and D. Peat, editors, Quantum Implications: Essays in Honour of David Bohm, pages 235-248. Law Book Co of Australasia, 1987.
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998.
Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(1):141-172, 2009. Preliminary version appeared in STOC '07.
Tsuyoshi Ito. Polynomial-space approximation of no-signaling provers. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming, ICALP '10, pages 140-151, 2010.
Tsuyoshi Ito, Hirotada Kobayashi, and Keiji Matsumoto. Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proceedings of the 24th IEEE Annual Conference on Computational Complexity, CCC '09, pages 217-228, 2009.
Tsuyoshi Ito and Thomas Vidick. A multi-prover interactive proof for NEXP sound against entangled provers. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science, FOCS '12, pages 243-252, 2012.
Nick S. Jones and Lluís Masanes. Interconversion of nonlocal correlations. Physical Review A, 72:052312, 2005.
Yael Kalai, Ran Raz, and Ron Rothblum. Delegation for bounded space. In Proceedings of the 45th ACM Symposium on the Theory of Computing, STOC '13, pages 565-574, 2013.
Yael Tauman Kalai, Ran Raz, and Oded Regev. On the space complexity of linear programming with preprocessing. In Proceedings of the 7th Innovations in Theoretical Computer Science Conference, ITCS '16, pages 293-300, 2016.
Yael Tauman Kalai, Ran Raz, and Ron D. Rothblum. How to delegate computations: the power of no-signaling proofs. In Proceedings of the 46th ACM Symposium on Theory of Computing, STOC '14, pages 485-494, 2014. Full version available at URL: https://eccc.weizmann.ac.il/report/2013/183/.
https://eccc.weizmann.ac.il/report/2013/183/
Leonid A. Khalfin and Boris S. Tsirelson. Quantum and quasi-classical analogs of Bell inequalities. Symposium on the Foundations of Modern Physics, pages 441-460, 1985.
Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter. Quantum nonlocality and beyond: Limits from nonlocal computation. Physical Review Letters, 99:180502, 2007.
Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992.
Lluís Masanes, Antonio Acín, and Nicolas Gisin. General properties of nonsignaling theories. Physical Review A, 73:012112, 2006.
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24(3):379-385, 1994.
Sandu Popescu and Daniel Rohrlich. Causality and Nonlocality as Axioms for Quantum Mechanics, pages 383-389. Springer Netherlands, 1998.
Prasad Raghavendra and David Steurer. Integrality gaps for strong SDP relaxations of UNIQUE GAMES. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, FOCS '09, pages 575-585, 2009. Full version at URL: http://people.eecs.berkeley.edu/~prasad/Files/cspgaps.pdf.
http://people.eecs.berkeley.edu/~prasad/Files/cspgaps.pdf
Peter Rastall. Locality, Bell’s theorem, and quantum mechanics. Foundations of Physics, 15(9):963-972, 1985.
Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th ACM Symposium on Theory of Computing, STOC '97, pages 475-484, 1997.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996.
Adi Shamir. IP = PSPACE. Journal of the ACM, 39(4):869-877, 1992.
Hanif D. Sherali and Warren P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3(3):411-430, 1990.
Anthony J. Short, Nicolas Gisin, and Sandu Popescu. The physics of no-bit-commitment: Generalized quantum non-locality versus oblivious transfer. Quantum Information Processing, 5(2):131-138, 2006.
Anthony J. Short, Sandu Popescu, and Nicolas Gisin. Entanglement swapping for generalized nonlocal correlations. Physical Review A, 73:012101, 2006.
Amir Shpilka and Avi Wigderson. Derandomizing homomorphism testing in general groups. In Proceedings of the 36th ACM Symposium on the Theory of Computing, STOC '04, pages 427-435, 2004.
Wim van Dam. Implausible consequences of superstrong nonlocality. Natural Computing, 12:9-12, 2013.
Thomas Vidick. Linearity testing with entangled provers, 2014. URL: http://users.cms.caltech.edu/~vidick/linearity_test.pdf.
http://users.cms.caltech.edu/~vidick/linearity_test.pdf
Stefan Wolf and Jürg Wullschleger. Oblivious transfer and quantum non-locality. In Proceedings of the 2005 International Symposium on Information Theory, ISIT '05, pages 1745-1748, 2005.
Alessandro Chiesa, Peter Manohar, and Igor Shinkar
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Earthmover Resilience and Testing in Ordered Structures
One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general.
In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images.
A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries.
characterizations of testability
distance estimation
earthmover resilient
ordered structures
property testing
Theory of computation~Streaming, sublinear and near linear time algorithms
18:1-18:35
Regular Paper
https://arxiv.org/abs/1801.09798
Omri
Ben-Eliezer
Omri Ben-Eliezer
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Eldar
Fischer
Eldar Fischer
Faculty of Computer Science, Israel Institute of Technology (Technion), Haifa, Israel
10.4230/LIPIcs.CCC.2018.18
Noga Alon and Omri Ben-Eliezer. Efficient removal lemmas for matrices. In Klaus Jansen, José D. P. Rolim, David Williamson, and Santosh Srinivas Vempala, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, volume 81 of LIPIcs, pages 25:1-25:18. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.25.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.25
Noga Alon, Omri Ben-Eliezer, and Eldar Fischer. Testing hereditary properties of ordered graphs and matrices. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 848-858. IEEE Computer Society, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.83.
http://dx.doi.org/10.1109/FOCS.2017.83
Noga Alon, Eldar Fischer, and Ilan Newman. Efficient testing of bipartite graphs for forbidden induced subgraphs. SIAM J. Comput., 37:959-976, 2007.
Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: It’s all about regularity. SIAM J. Comput., 39(1):143-167, 2009. URL: http://dx.doi.org/10.1137/060667177.
http://dx.doi.org/10.1137/060667177
Noga Alon and Asaf Shapira. A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comput., 37(6):1703-1727, 2008. URL: http://dx.doi.org/10.1137/06064888X.
http://dx.doi.org/10.1137/06064888X
Barry C. Arnold, N. Balakrishnan, and H. N. Nagaraja. A First Course in Order Statistics (Classics in Applied Mathematics). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
Tim Austin and Terence Tao. Testability and repair of hereditary hypergraph properties. Random Struct. Algorithms, 36:373-463, 2010.
Maria Axenovich and Ryan R. Martin. A version of Szemerédi’s regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs. arXiv, 1106:2871, 2011.
Omri Ben-Eliezer, Simon Korman, and Daniel Reichman. Deleting and testing forbidden patterns in multi-dimensional arrays. 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 9:1-9:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.9.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.9
Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. The power and limitations of uniform samples in testing properties of figures. 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 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2016.45.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2016.45
Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Tolerant testers of image properties. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 90:1-90:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.90.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.90
Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 164-173. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591887.
http://dx.doi.org/10.1145/2591796.2591887
Eric Blais and Yuichi Yoshida. A characterization of constant-sample testable properties. arXiv, 1612:06016, 2016.
Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, Balázs Szegedy, and Katalin Vesztergombi. Graph limits and parameter testing. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 261-270, New York, NY, USA, 2006. ACM.
Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-Monotonicity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 29:1-29:21, 2017.
Xi Chen, Adam Freilich, Rocco A. Servedio, and Timothy Sun. Sample-based high-dimensional convexity testing. In Klaus Jansen, José D. P. Rolim, David Williamson, and Santosh Srinivas Vempala, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, volume 81 of LIPIcs, pages 37:1-37:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.37.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.37
Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Dorit S. Hochbaum, Klaus Jansen, José D. P. Rolim, and Alistair Sinclair, editors, Randomization, Approximation, and Combinatorial Algorithms and Techniques, Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems RANDOM-APPROX'99, Berkeley, CA, USA, August 8-11, 1999, Proceedings, volume 1671 of Lecture Notes in Computer Science, pages 97-108. Springer, 1999. URL: http://dx.doi.org/10.1007/978-3-540-48413-4_10.
http://dx.doi.org/10.1007/978-3-540-48413-4_10
Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. J. Comput. Syst. Sci., 60(3):717-751, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1692.
http://dx.doi.org/10.1006/jcss.1999.1692
Eldar Fischer. Testing graphs for colorability properties. Random Struct. Algorithms, 26(3):289-309, 2005. URL: http://dx.doi.org/10.1002/rsa.20037.
http://dx.doi.org/10.1002/rsa.20037
Eldar Fischer and Lance Fortnow. Tolerant versus intolerant testing for boolean properties. Theory of Computing, 2(9):173-183, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a009.
http://dx.doi.org/10.4086/toc.2006.v002a009
Eldar Fischer and Ilan Newman. Testing of matrix-poset properties. Combinatorica, 27(3):293-327, 2007. URL: http://dx.doi.org/10.1007/s00493-007-2154-3.
http://dx.doi.org/10.1007/s00493-007-2154-3
Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM J. Comput., 37(2):482-501, 2007. URL: http://dx.doi.org/10.1137/060652324.
http://dx.doi.org/10.1137/060652324
Eldar Fischer and Eyal Rozenberg. Lower bounds for testing forbidden induced substructures in bipartite-graph-like combinatorial objects. In Moses Charikar, Klaus Jansen, Omer Reingold, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 10th International Workshop, APPROX 2007, and 11th International Workshop, RANDOM 2007, Princeton, NJ, USA, August 20-22, 2007, Proceedings, volume 4627 of Lecture Notes in Computer Science, pages 464-478. Springer, 2007. URL: http://dx.doi.org/10.1007/978-3-540-74208-1_34.
http://dx.doi.org/10.1007/978-3-540-74208-1_34
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. URL: http://dx.doi.org/10.1145/285055.285060.
http://dx.doi.org/10.1145/285055.285060
Oded Goldreich and Madhu Sudan. Locally testable codes and pcps of almost-linear length. J. ACM, 53(4):558-655, 2006. URL: http://dx.doi.org/10.1145/1162349.1162351.
http://dx.doi.org/10.1145/1162349.1162351
Oded Goldreich and Luca Trevisan. Three theorems regarding testing graph properties. Random Struct. Algorithms, 23(1):23-57, 2003. URL: http://dx.doi.org/10.1002/rsa.10078.
http://dx.doi.org/10.1002/rsa.10078
Carlos Hoppen, Yoshiharu Kohayakawa, Richard Lang, Hanno Lefmann, and Henrique Stagni. Estimating parameters associated with monotone properties. In Klaus Jansen, Claire Mathieu, José D. P. Rolim, and Chris Umans, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, September 7-9, 2016, Paris, France, volume 60 of LIPIcs, pages 35:1-35:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.35.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.35
Carlos Hoppen, Yoshiharu Kohayakawa, Richard Lang, Hanno Lefmann, and Henrique Stagni. Estimating the distance to a hereditary graph property. Electronic Notes in Discrete Mathematics, 61:607-613, 2017. URL: http://dx.doi.org/10.1016/j.endm.2017.07.014.
http://dx.doi.org/10.1016/j.endm.2017.07.014
Felix Joos, Jaehoon Kim, Daniela Kühn, and Deryk Osthus. A characterization of testable hypergraph properties. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 859-867. IEEE Computer Society, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.84.
http://dx.doi.org/10.1109/FOCS.2017.84
László Lovász and Balázs Szegedy. Testing properties of graphs and functions. Israel Journal of Mathematics, 178:113-156, 2010.
Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012-1042, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.03.002.
http://dx.doi.org/10.1016/j.jcss.2006.03.002
Sofya Raskhodnikova. Approximate testing of visual properties. In Sanjeev Arora, Klaus Jansen, José D. P. Rolim, and Amit Sahai, editors, Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2003 and 7th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2003, Princeton, NJ, USA, August 24-26, 2003, Proceedings, volume 2764 of Lecture Notes in Computer Science, pages 370-381. Springer, 2003. URL: http://dx.doi.org/10.1007/978-3-540-45198-3_31.
http://dx.doi.org/10.1007/978-3-540-45198-3_31
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996. URL: http://dx.doi.org/10.1137/S0097539793255151.
http://dx.doi.org/10.1137/S0097539793255151
Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99-121, 2000. URL: http://dx.doi.org/10.1023/A:1026543900054.
http://dx.doi.org/10.1023/A:1026543900054
Endre Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399-401. CNRS, Paris, 1978.
Omri Ben-Eliezer and Eldar Fischer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
New Hardness Results for the Permanent Using Linear Optics
In 2011, Aaronson gave a striking proof, based on quantum linear optics, that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant's seminal proof of the same fact in 1979. Nevertheless, it did not show #P-hardness of the permanent for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques.
First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson's original proof, this implies that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents of orthogonal matrices over the finite field F_{p^4} for p != 2, 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan.
Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix. This result shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly, despite the fact that they can be efficiently sampled by a classical computer.
Permanent
Linear optics
#P-hardness
Orthogonal matrices
Theory of computation~Problems, reductions and completeness
19:1-19:29
Regular Paper
Both authors were supported by the Vannevar Bush Faculty Fellowship from the US Department of Defense. Part of this research was completed while visiting UT Austin.
Daniel
Grier
Daniel Grier
MIT, Cambridge, USA
Supported by an NSF Graduate Research Fellowship under Grant No. 1122374.
Luke
Schaeffer
Luke Schaeffer
MIT, Cambridge, USA
10.4230/LIPIcs.CCC.2018.19
S. Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proc. Roy. Soc. London, A461(2063):3473-3482, 2005. quant-ph/0412187.
S. Aaronson. A linear-optical proof that the permanent is #P-hard. Proc. Roy. Soc. London, A467(2088):3393-3405, 2011. arXiv:1109.1674.
S. Aaronson and A. Arkhipov. The computational complexity of linear optics. Theory of Computing, 9(4):143-252, 2013. Conference version in Proceedings of ACM STOC'2011. ECCC TR10-170, arXiv:1011.3245.
S. Aaronson, D. Grier, and L. Schaeffer. The classification of reversible bit operations. arXiv preprint arXiv:1504.05155, 2015.
Leonard M. Adleman, Jonathan DeMarrais, and Ming-Deh A. Huang. Quantum computability. SIAM J. Comput., 26(5):1524-1540, 1997. URL: http://dx.doi.org/10.1137/S0097539795293639.
http://dx.doi.org/10.1137/S0097539795293639
A. Ben-Dor and S. Halevi. Zero-one permanent is #P-complete, a simpler proof. In ISTCS, pages 108-117, 1993.
S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Information processing letters, 18(3):147-150, 1984.
M. Bremner, R. Jozsa, and D. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. Roy. Soc. London, A467(2126):459-472, 2010. arXiv:1005.1407.
E. R. Caianiello. On quantum field theory, 1: explicit solution of Dyson’s equation in electrodynamics without use of Feynman graphs. Nuovo Cimento, 10:1634-1652, 1953.
L. Chakhmakhchyan, N. J. Cerf, and R. Garcia-Patron. A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices. arXiv preprint arXiv:1609.02416, 2016.
A. Drucker and R. de Wolf. Quantum proofs for classical theorems. Theory of Computing Graduate Surveys, pages 1-54, 2011. arXiv:0910.3376, ECCC TR03-048.
S. Fenner, F. Green, S. Homer, and R. Pruim. Quantum NP is hard for PH. In Proceedings of 6th Italian Conference on theoretical Computer Science, pages 241-252. Citeseer, 1998.
Stephen A. Fenner, Lance Fortnow, and Stuart A. Kurtz. Gap-definable counting classes. J. Comput. Syst. Sci., 48(1):116-148, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80024-8.
http://dx.doi.org/10.1016/S0022-0000(05)80024-8
Leonid Gurvits. On the complexity of mixed discriminants and related problems. In Joanna Jedrzejowicz and Andrzej Szepietowski, editors, Mathematical Foundations of Computer Science 2005, 30th International Symposium, MFCS 2005, Gdansk, Poland, August 29 - September 2, 2005, Proceedings, volume 3618 of Lecture Notes in Computer Science, pages 447-458. Springer, 2005. URL: http://dx.doi.org/10.1007/11549345_39.
http://dx.doi.org/10.1007/11549345_39
C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett., 59(18):2044-2046, 1987.
E. Knill. Quantum gates using linear optics and postselection. Physical Review A, 66(5), 2002. URL: http://dx.doi.org/10.1103/PhysRevA.66.052306.
http://dx.doi.org/10.1103/PhysRevA.66.052306
E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409:46-52, 2001. See also quant-ph/0006088.
G. Kogan. Computing permanents over fields of characteristic 3: Where and why it becomes difficult. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 108-114. IEEE, 1996.
G. Kuperberg. How hard is it to approximate the Jones polynomial? Theory of Computing, 11(6):183-219, 2015.
M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.
S. Rahimi-Keshari, A. P. Lund, and T. C. Ralph. What can quantum optics say about computational complexity theory? Physical review letters, 114(6):060501, 2015.
Terry Rudolph. Simple encoding of a quantum circuit amplitude as a matrix permanent. Physical Review A, 80(5):054302, 2009.
Rimli Sengupta. Cancellation is exponentially powerful for computing the determinant. Information Processing Letters, 62(4):177-181, 1997.
Y. Shi. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Information and Computation, 3(1):84-92, 2002. quant-ph/0205115.
L. J. Stockmeyer. The complexity of approximate counting. In Proc. ACM STOC, pages 118-126, 1983.
S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991.
Seinosuke Toda and Mitsunori Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM J. Comput., 21(2):316-328, 1992. URL: http://dx.doi.org/10.1137/0221023.
http://dx.doi.org/10.1137/0221023
T. Toffoli. Reversible computing. In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 632-644. Springer, 1980.
L. Troyansky and N. Tishby. Permanent uncertainty: On the quantum evaluation of the determinant and the permanent of a matrix. In Proceedings of PhysComp, 1996.
N. Tschebotareff. Die bestimmung der dichtigkeit einer menge von primzahlen, welche zu einer gegebenen substitutionsklasse gehören. Mathematische Annalen, 95(1):191-228, 1926. URL: http://dx.doi.org/10.1007/BF01206606.
http://dx.doi.org/10.1007/BF01206606
L. G. Valiant. The complexity of computing the permanent. Theoretical Comput. Sci., 8(2):189-201, 1979.
Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL: http://dx.doi.org/10.1145/800135.804419.
http://dx.doi.org/10.1145/800135.804419
Leslie G. Valiant. Negation can be exponentially powerful. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 189-196. ACM, 1979. URL: http://dx.doi.org/10.1145/800135.804412.
http://dx.doi.org/10.1145/800135.804412
Daniel Grier and Luke Schaeffer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Retracted: Two-Player Entangled Games are NP-Hard
The article, published on June 4th, 2018 in the CCC 2018 proceedings, has been retracted by agreement between the authors, the editor(s), and the publisher Schloss Dagstuhl / LIPIcs. The retraction has been agreed due to an error in the proof of the main result. This error is carried over from an error in the referenced paper “Three-player entangled XOR games are NP-hard to approximate” by Thomas Vidick (SICOMP ’16). That paper was used in an essential way to obtain the present result, and the error cannot be addressed through an erratum. See Retraction Notice on the last page of the PDF.
We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP subseteq MIP^*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers.
low-degree testing
entangled nonlocal games
multi-prover interactive proof systems
Theory of computation~Quantum complexity theory
Theory of computation~Interactive proof systems
Theory of computation~Complexity classes
20:1-20:18
Regular Paper
Anand
Natarajan
Anand Natarajan
Center for Theoretical Physics, MIT, Cambridge, USA
https://orcid.org/0000-0003-3648-3844
Supported by NSF CAREER Grant CCF-1452616
Thomas
Vidick
Thomas Vidick
Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, USA
https://orcid.org/0000-0002-6405-365X
Supported by NSF CAREER Grant CCF-1553477, AFOSR YIP award number FA9550-16-1-0495, and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028).
10.4230/LIPIcs.CCC.2018.20
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998.
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998.
László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991.
John S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195-200, 1964.
Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47:549-595, 1993.
Richard Cleve, Peter Høyer, Ben Toner, and John Watrous. Consequences and limits of nonlocal strategies. 2004. URL: http://arxiv.org/abs/quant-ph/0404076.
http://arxiv.org/abs/quant-ph/0404076
Irit Dinur, David Steurer, and Thomas Vidick. A parallel repetition theorem for entangled projection games. Computational Complexity, 24(2):201-254, 2015. URL: http://dx.doi.org/10.1007/s00037-015-0098-3.
http://dx.doi.org/10.1007/s00037-015-0098-3
Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical review, 47(10):777, 1935.
Uriel Feige, Shafi Goldwasser, Laszlo Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996.
Joseph Fitzsimons and Thomas Vidick. A multiprover interactive proof system for the local Hamiltonian problem. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pages 103-112. ACM, 2015.
Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: Interactive proofs for muggles. Journal of the ACM (JACM), 62(4):27, 2015.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4):798-859, 2001.
Tsuyoshi Ito, Hirotada Kobayashi, and Keiji Matsumoto. Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proceedings: Twenty-Fourth Annual IEEE Conference on Computational Complexity (CCC 2009), pages 217-228, July 2009.
Tsuyoshi Ito and Thomas Vidick. A multi-prover interactive proof for NEXP sound against entangled provers. Proc. 53rd FOCS, pages 243-252, 2012. URL: http://arxiv.org/abs/1207.0550.
http://arxiv.org/abs/1207.0550
Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP = PSPACE. Communications of the ACM, 53(12):102-109, 2010. URL: http://arxiv.org/abs/0907.4737.
http://arxiv.org/abs/0907.4737
Zhengfeng Ji. Classical verification of quantum proofs. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 885-898. ACM, 2016.
Zhengfeng Ji. Compression of quantum multi-prover interactive proofs. 2016. URL: http://arxiv.org/abs/1610.03133.
http://arxiv.org/abs/1610.03133
Yael Tauman Kalai, Ran Raz, and Ron D. Rothblum. How to delegate computations: the power of no-signaling proofs. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 485-494. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591809.
http://dx.doi.org/10.1145/2591796.2591809
Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM (JACM), 39(4):859-868, 1992.
Anand Natarajan and Thomas Vidick. A quantum linearity test for robustly verifying entanglement. 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 1003-1015. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055468.
http://dx.doi.org/10.1145/3055399.3055468
Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP. 2018. URL: http://arxiv.org/abs/1801.03821v2.
http://arxiv.org/abs/1801.03821v2
Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 475-484. ACM, 1997. URL: http://dx.doi.org/10.1145/258533.258641.
http://dx.doi.org/10.1145/258533.258641
Ben Reichardt, Falk Unger, and Umesh Vazirani. A classical leash for a quantum system: Command of quantum systems via rigidity of CHSH games. Nature, 496(7446):456-460, 2013.
Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, 1992.
Ben Toner. Monogamy of non-local quantum correlations. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465(2101):59-69, 2009. URL: http://dx.doi.org/10.1098/rspa.2008.0149.
http://dx.doi.org/10.1098/rspa.2008.0149
Thomas Vidick. Three-player entangled XOR games are NP-hard to approximate. In Proc. 54th FOCS, 2013. URL: http://arxiv.org/abs/1302.1242.
http://arxiv.org/abs/1302.1242
Anand Natarajan and Thomas Vidick
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Complexity Classification of Conjugated Clifford Circuits
Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.
gate set classification
quantum advantage
sampling problems
polynomial hierarchy
Theory of computation~Quantum complexity theory
Theory of computation~Computational complexity and cryptography
21:1-21:25
Regular Paper
https://arxiv.org/abs/1709.01805
Adam
Bouland
Adam Bouland
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
https://orcid.org/0000-0002-8556-8337
AB was partially supported by the NSF GRFP under Grant No. 1122374, by a Vannevar Bush Fellowship from the US Department of Defense, and by an NSF Waterman award under grant number 1249349.
Joseph F.
Fitzsimons
Joseph F. Fitzsimons
Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 , Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
JFF acknowledges support from the Air Force Office of Scientific Research under AOARD grant no. FA2386-15-1-4082. This material is based on research supported in part by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2013-01.
Dax Enshan
Koh
Dax Enshan Koh
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
https://orcid.org/0000-0002-8968-591X
DEK is supported by the National Science Scholarship from the Agency for Science, Technology and Research (A*STAR).
10.4230/LIPIcs.CCC.2018.21
Universal sets of gates for SU(3)?, 2012. Accessed: 2017-08-01. URL: https://cstheory.stackexchange.com/questions/11308/universal-sets-of-gates-for-su3.
https://cstheory.stackexchange.com/questions/11308/universal-sets-of-gates-for-su3
Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 461(2063):3473-3482, 2005. URL: http://dx.doi.org/10.1098/rspa.2005.1546.
http://dx.doi.org/10.1098/rspa.2005.1546
Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In Proceedings of the forty-third annual ACM Symposium on Theory of Computing, pages 333-342. ACM, 2011.
Scott Aaronson and Lijie Chen. Complexity-theoretic foundations of quantum supremacy experiments. Proc. CCC, 2017.
Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70(5):052328, 2004.
Scott Aaronson, Daniel Grier, and Luke Schaeffer. The classification of reversible bit operations. In Proceedings of Innovations in Theoretical Computer Science (ITCS), 2017.
Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error. In Proceedings of the twenty-ninth annual ACM Symposium on Theory of Computing, pages 176-188. ACM, 1997.
Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. arXiv:1608.00263, 2016.
Adam Bouland and Scott Aaronson. Generation of universal linear optics by any beam splitter. Physical Review A, 89(6):062316, 2014.
Adam Bouland, Laura Mancinska, and Xue Zhang. Complexity classification of two-qubit commuting hamiltonians. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 28:1-28:33. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.28.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.28
Sergey Bravyi, David Gosset, and Robert Koenig. Quantum advantage with shallow circuits. arXiv:1704.00690, 2017.
Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Physical Review A, 86(5):052329, 2012.
Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71(2):022316, 2005.
Michael J Bremner, Richard Jozsa, and Dan J Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, page rspa20100301. The Royal Society, 2010.
Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. Physical Review Letters, 117(8):080501, 2016.
Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum, 1, 2017.
Hans J Briegel, David E Browne, W Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5(1):19-26, 2009.
Christopher M Dawson and Michael A Nielsen. The Solovay-Kitaev algorithm. Quantum Information &Computation, 6(1):81-95, 2006.
David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with efficient quantum codes. Phys. Rev. Lett., 77:3260-3263, Oct 1996. URL: http://dx.doi.org/10.1103/PhysRevLett.77.3260.
http://dx.doi.org/10.1103/PhysRevLett.77.3260
Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Physical Review Letters, 102(11):110502, 2009.
Edward Farhi and Aram W Harrow. Quantum supremacy through the quantum approximate optimization algorithm. arXiv:1602.07674, 2016.
Bill Fefferman and Christopher Umans. On the power of quantum fourier sampling. In Anne Broadbent, editor, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2016, September 27-29, 2016, Berlin, Germany, volume 61 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.TQC.2016.1.
http://dx.doi.org/10.4230/LIPIcs.TQC.2016.1
Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Impossibility of classically simulating one-clean-qubit computation. arXiv:1409.6777, 2014.
Daniel Gottesman. Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology, arXiv:quant-ph/9705052, 1997.
Daniel Gottesman. The Heisenberg representation of quantum computers. Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pages 32-43, 1999.
Chris Granade and Ben Criger. QuaEC: Quantum error correction analysis in Python. http://www.cgranade.com/python-quaec/groups.html#, 2012. Accessed: 2017-06-01.
http://www.cgranade.com/python-quaec/groups.html#
Daniel Grier and Luke Schaeffer. The classification of stabilizer operations over qubits. arXiv:1603.03999, 2016.
Amihay Hanany and Yang-Hui He. A monograph on the classification of the discrete subgroups of SU(4). Journal of High Energy Physics, 2001(02):027, 2001.
Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert. Anti-concentration theorems for schemes showing a quantum computational supremacy. arXiv:1706.03786, 2017.
Daniel Harlow. Jerusalem lectures on black holes and quantum information. Reviews of Modern Physics, 88(1):015002, 2016.
Aram Harrow and Saeed Mehraban. Personal communication, 2018.
Richard Jozsa and Akimasa Miyake. Matchgates and classical simulation of quantum circuits. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 464(2100):3089-3106, 2008. URL: http://dx.doi.org/10.1098/rspa.2008.0189.
http://dx.doi.org/10.1098/rspa.2008.0189
Richard Jozsa and Maarten Van den Nest. Classical simulation complexity of extended Clifford circuits. Quantum Information and Computation, 14(7/8):633-648, 2014.
Dax Enshan Koh. Further extensions of Clifford circuits and their classical simulation complexities. Quantum Information &Computation, 17(3&4):0262-0282, 2017.
Greg Kuperberg. How hard is it to approximate the Jones polynomial? Theory of Computing, 11(6):183-219, 2015.
Richard E. Ladner. On the structure of polynomial time reducibility. J. ACM, 22(1):155-171, 1975.
Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Physical Review Letters, 77(1):198, 1996.
Easwar Magesan, Jay M Gambetta, and Joseph Emerson. Scalable and robust randomized benchmarking of quantum processes. Physical Review Letters, 106(18):180504, 2011.
Ryan L. Mann and Michael J. Bremner. On the complexity of random quantum computations and the Jones polynomial. arXiv:1711.00686, 2017.
Enrique Martin-Lopez, Anthony Laing, Thomas Lawson, Roberto Alvarez, Xiao-Qi Zhou, and Jeremy L O'Brien. Experimental realization of shor’s quantum factoring algorithm using qubit recycling. Nature Photonics, 6(11):773-776, 2012.
Dmitri Maslov and Martin Roetteler. Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations. arXiv:1705.09176, 2017.
Tomoyuki Morimae. Hardness of classically sampling one clean qubit model with constant total variation distance error. arXiv:1704.03640, 2017.
Tomoyuki Morimae, Keisuke Fujii, and Joseph F Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Physical Review Letters, 112(13):130502, 2014.
Gabriele Nebe, Eric M Rains, and Neil JA Sloane. The invariants of the clifford groups. Designs, Codes and Cryptography, 24(1):99-122, 2001.
Gabriele Nebe, Eric M Rains, and Neil James Alexander Sloane. Self-dual codes and invariant theory, volume 17. Springer, 2006.
Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information, 2002.
Michał Oszmaniec and Zoltán Zimborás. Universal extensions of restricted classes of quantum operations. arXiv:1705.11188, 2017.
John Preskill. Quantum computing and the entanglement frontier. arXiv:1203.5813, 2012.
Robert Raussendorf and Hans J Briegel. A one-way quantum computer. Physical Review Letters, 86(22):5188, 2001.
Robert Raussendorf, Daniel E Browne, and Hans J Briegel. Measurement-based quantum computation on cluster states. Physical Review A, 68(2):022312, 2003.
Imdad SB Sardharwalla, Toby S Cubitt, Aram W Harrow, and Noah Linden. Universal refocusing of systematic quantum noise. arXiv:1602.07963, 2016.
Adam Sawicki and Katarzyna Karnas. Criteria for universality of quantum gates. Phys. Rev. A, 95:062303, Jun 2017.
Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41(2):303-332, 1999.
Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 452(1954):2551-2577, 1996. URL: http://dx.doi.org/10.1098/rspa.1996.0136.
http://dx.doi.org/10.1098/rspa.1996.0136
Larry Stockmeyer. The complexity of approximate counting. In Proceedings of the fifteenth annual ACM Symposium on Theory of Computing, pages 118-126. ACM, 1983.
Barbara M Terhal and David P DiVincenzo. Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. Quantum Information &Computation, 4(2):134-145, 2004.
Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. URL: http://dx.doi.org/10.1137/0220053.
http://dx.doi.org/10.1137/0220053
Zak Webb. The Clifford group forms a unitary 3-design. Quantum Information and Computation, 16:1379-1400, 2016.
Huangjun Zhu. Multiqubit clifford groups are unitary 3-designs. Physical Review A, 96(6):062336, 2017.
Adam Bouland, Joseph F. Fitzsimons, and Dax E. Koh
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Efficient Batch Verification for UP
Consider a setting in which a prover wants to convince a verifier of the correctness of k NP statements. For example, the prover wants to convince the verifier that k given integers N_1,...,N_k are all RSA moduli (i.e., products of equal length primes). Clearly this problem can be solved by simply having the prover send the k NP witnesses, but this involves a lot of communication. Can interaction help? In particular, is it possible to construct interactive proofs for this task whose communication grows sub-linearly with k?
Our main result is such an interactive proof for verifying the correctness of any k UP statements (i.e., NP statements that have a unique witness). The proof-system uses only a constant number of rounds and the communication complexity is k^delta * poly(m), where delta>0 is an arbitrarily small constant, m is the length of a single witness, and the poly term refers to a fixed polynomial that only depends on the language and not on delta. The (honest) prover strategy can be implemented in polynomial-time given access to the k (unique) witnesses.
Our proof leverages "interactive witness verification" (IWV), a new type of proof-system that may be of independent interest. An IWV is a proof-system in which the verifier needs to verify the correctness of an NP statement using: (i) a sublinear number of queries to an alleged NP witness, and (ii) a short interaction with a powerful but untrusted prover. In contrast to the setting of PCPs and Interactive PCPs, here the verifier only has access to the raw NP witness, rather than some encoding thereof.
Interactive Proof
Batch Verification
Unique Solution
Theory of computation~Computational complexity and cryptography
22:1-22:23
Regular Paper
https://eccc.weizmann.ac.il/report/2018/022
Omer
Reingold
Omer Reingold
Stanford University, Palo Alto CA, USA
Supported by NSF grant CCF-1749750.
Guy N.
Rothblum
Guy N. Rothblum
Weizmann Institute, Rehovot, Israel
Ron D.
Rothblum
Ron D. Rothblum
MIT and Northeastern University, Cambridge and Boston MA, USA
Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship and in part by the Defense Advanced Research Projects Agency (DARPA), the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236 and by the Cybersecurity and Privacy Institute at Northeastern University.
10.4230/LIPIcs.CCC.2018.22
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
http://dx.doi.org/10.1145/278298.278306
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
http://dx.doi.org/10.1145/278298.278306
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs; A new characterization of NP. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 2-13. IEEE Computer Society, 1992. URL: http://dx.doi.org/10.1109/SFCS.1992.267824.
http://dx.doi.org/10.1109/SFCS.1992.267824
László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 21-31. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103428.
http://dx.doi.org/10.1145/103418.103428
László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. URL: http://dx.doi.org/10.1007/BF01200056.
http://dx.doi.org/10.1007/BF01200056
László Babai and Shlomo Moran. Arthur-merlin games: A randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci., 36(2):254-276, 1988. URL: http://dx.doi.org/10.1016/0022-0000(88)90028-1.
http://dx.doi.org/10.1016/0022-0000(88)90028-1
Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 113-131. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62223.
http://dx.doi.org/10.1145/62212.62223
Eli Ben-Sasson, Alessandro Chiesa, and Nicholas Spooner. Interactive oracle proofs. Cryptology ePrint Archive, Report 2016/116, 2016. URL: http://eprint.iacr.org/.
http://eprint.iacr.org/
Itay Berman, Ron D. Rothblum, and Vinod Vaikuntanathan. Zero-knowledge proofs of proximity. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 19:1-19:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.19.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.19
Zvika Brakerski, Justin Holmgren, and Yael Tauman Kalai. Non-interactive delegation and batch NP verification from standard computational assumptions. 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 474-482. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055497.
http://dx.doi.org/10.1145/3055399.3055497
Alessandro Chiesa and Tom Gur. Proofs of proximity for distribution testing. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 53:1-53:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.53.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.53
Funda Ergün, Ravi Kumar, and Ronitt Rubinfeld. Fast approximate probabilistically checkable proofs. Inf. Comput., 189(2):135-159, 2004. URL: http://dx.doi.org/10.1016/j.ic.2003.09.005.
http://dx.doi.org/10.1016/j.ic.2003.09.005
Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996. URL: http://dx.doi.org/10.1145/226643.226652.
http://dx.doi.org/10.1145/226643.226652
Eldar Fischer, Yonatan Goldhirsh, and Oded Lachish. Partial tests, universal tests and decomposability. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 483-500. ACM, 2014. URL: http://dx.doi.org/10.1145/2554797.2554841.
http://dx.doi.org/10.1145/2554797.2554841
Lance Fortnow, John Rompel, and Michael Sipser. On the power of multi-prover interactive protocols. Theor. Comput. Sci., 134(2):545-557, 1994. URL: http://dx.doi.org/10.1016/0304-3975(94)90251-8.
http://dx.doi.org/10.1016/0304-3975(94)90251-8
Martin Fürer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. Advances in Computing Research, 5:429-442, 1989.
Oded Goldreich. Modern cryptography, probabilistic proofs and pseudorandomness, volume 17 of Algorithms and Combinatorics. Springer-Verlag, 1999.
Oded Goldreich. Overview of the doubly-efficient interactive proof systems of RRR. Electronic Colloquium on Computational Complexity (ECCC), 24:102, 2017. URL: https://eccc.weizmann.ac.il/report/2017/102.
https://eccc.weizmann.ac.il/report/2017/102
Oded Goldreich, Tom Gur, and Ron D. Rothblum. Proofs of proximity for context-free languages and read-once branching programs - (extended abstract). 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 666-677. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_54.
http://dx.doi.org/10.1007/978-3-662-47672-7_54
Oded Goldreich and Johan Håstad. On the complexity of interactive proofs with bounded communication. Inf. Process. Lett., 67(4):205-214, 1998. URL: http://dx.doi.org/10.1016/S0020-0190(98)00116-1.
http://dx.doi.org/10.1016/S0020-0190(98)00116-1
Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. J. ACM, 38(3):691-729, 1991. URL: http://dx.doi.org/10.1145/116825.116852.
http://dx.doi.org/10.1145/116825.116852
Oded Goldreich, Salil P. Vadhan, and Avi Wigderson. On interactive proofs with a laconic prover. Computational Complexity, 11(1-2):1-53, 2002. URL: http://dx.doi.org/10.1007/s00037-002-0169-0.
http://dx.doi.org/10.1007/s00037-002-0169-0
Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: interactive proofs for muggles. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 113-122. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374396.
http://dx.doi.org/10.1145/1374376.1374396
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM J. Comput., 18(1):186-208, 1989. URL: http://dx.doi.org/10.1137/0218012.
http://dx.doi.org/10.1137/0218012
Tom Gur and Ron D. Rothblum. Non-interactive proofs of proximity. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 133-142. ACM, 2015. URL: http://dx.doi.org/10.1145/2688073.2688079.
http://dx.doi.org/10.1145/2688073.2688079
Tom Gur and Ron D. Rothblum. A hierarchy theorem for interactive proofs of proximity. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 39:1-39:43. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.39.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.39
Yael Tauman Kalai and Ran Raz. Probabilistically checkable arguments. In Shai Halevi, editor, Advances in Cryptology - CRYPTO 2009, 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2009. Proceedings, volume 5677 of Lecture Notes in Computer Science, pages 143-159. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03356-8_9.
http://dx.doi.org/10.1007/978-3-642-03356-8_9
Yael Tauman Kalai and Ron D. Rothblum. Arguments of proximity - [extended abstract]. In Rosario Gennaro and Matthew Robshaw, editors, Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2015, Proceedings, Part II, volume 9216 of Lecture Notes in Computer Science, pages 422-442. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48000-7_21.
http://dx.doi.org/10.1007/978-3-662-48000-7_21
Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. URL: http://dx.doi.org/10.1145/146585.146605.
http://dx.doi.org/10.1145/146585.146605
Omer Reingold, Guy N. Rothblum, and Ron Rothblum. Efficient batch verification for UP. Electronic Colloquium on Computational Complexity (ECCC), 25:22, 2018. URL: https://eccc.weizmann.ac.il/report/2018/022.
https://eccc.weizmann.ac.il/report/2018/022
Omer Reingold, Guy N. Rothblum, and Ron D. Rothblum. Constant-round interactive proofs for delegating computation. 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 49-62. ACM, 2016. URL: http://dx.doi.org/10.1145/2897518.2897652.
http://dx.doi.org/10.1145/2897518.2897652
Guy N. Rothblum, Salil P. Vadhan, and Avi Wigderson. Interactive proofs of proximity: delegating computation in sublinear time. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 793-802. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488709.
http://dx.doi.org/10.1145/2488608.2488709
Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, 1992. URL: http://dx.doi.org/10.1145/146585.146609.
http://dx.doi.org/10.1145/146585.146609
L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.
Omer Reingold, Guy N. Rothblum, and Ron D. Rothblum
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Tight Lower Bound for Entropy Flattening
We study entropy flattening: Given a circuit C_X implicitly describing an n-bit source X (namely, X is the output of C_X on a uniform random input), construct another circuit C_Y describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have C_Y evaluate C_X altogether Theta(n^2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit C_Y for entropy flattening that repeatedly queries C_X as an oracle requires Omega(n^2) queries.
Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [Johan Håstad et al., 1999; John Rompel, 1990; Thomas Holenstein, 2006; Iftach Haitner et al., 2006; Iftach Haitner et al., 2009; Iftach Haitner et al., 2013; Iftach Haitner et al., 2010; Salil P. Vadhan and Colin Jia Zheng, 2012]. It is also used in reductions between problems complete for statistical zero-knowledge [Tatsuaki Okamoto, 2000; Amit Sahai and Salil P. Vadhan, 1997; Oded Goldreich et al., 1999; Vadhan, 1999]. The Theta(n^2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.
Entropy
One-way function
Theory of computation~Computational complexity and cryptography
23:1-23:28
Regular Paper
Yi-Hsiu
Chen
Yi-Hsiu Chen
School of Engineering and Applied Sciences, Harvard University, USA
Supported by NSF grant CCF-1749750
Mika
Göös
Mika Göös
School of Engineering and Applied Sciences, Harvard University, USA
Supported by Michael O. Rabin Postdoctoral Fellowship
Salil P.
Vadhan
Salil P. Vadhan
Computer Science and Applied Mathematics, Harvard University, USA
Supported by NSF grant CCF-1749750
Jiapeng
Zhang
Jiapeng Zhang
University of California San Diego, USA
Supported by NSF CCF-1614023
10.4230/LIPIcs.CCC.2018.23
Yevgeniy Dodis, Rafail Ostrovsky, Leonid Reyzin, and Adam D. Smith. Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. SIAM J. Comput., 38(1):97-139, 2008. URL: http://dx.doi.org/10.1137/060651380.
http://dx.doi.org/10.1137/060651380
Rosario Gennaro, Yael Gertner, Jonathan Katz, and Luca Trevisan. Bounds on the efficiency of generic cryptographic constructions. SIAM J. Comput., 35(1):217-246, 2005. URL: http://dx.doi.org/10.1137/S0097539704443276.
http://dx.doi.org/10.1137/S0097539704443276
Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In David S. Johnson, editor, Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, Seattle, Washigton, USA, pages 25-32. ACM, 1989. URL: http://dx.doi.org/10.1145/73007.73010.
http://dx.doi.org/10.1145/73007.73010
Oded Goldreich, Amit Sahai, and Salil P. Vadhan. Can statistical zero knowledge be made non-interactive? or on the relationship of SZK and NISZK. In Michael J. Wiener, editor, Advances in Cryptology - CRYPTO '99, 19th Annual International Cryptology Conference, Santa Barbara, California, USA, August 15-19, 1999, Proceedings, volume 1666 of Lecture Notes in Computer Science, pages 467-484. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48405-1_30.
http://dx.doi.org/10.1007/3-540-48405-1_30
Oded Goldreich, Amit Sahai, and Salil P. Vadhan. Can statistical zero knowledge be made non-interactive? or on the relationship of SZK and NISZK. Electronic Colloquium on Computational Complexity (ECCC), 6(13), 1999. URL: http://eccc.hpi-web.de/eccc-reports/1999/TR99-013/index.html.
http://eccc.hpi-web.de/eccc-reports/1999/TR99-013/index.html
Iftach Haitner, Danny Harnik, and Omer Reingold. Efficient pseudorandom generators from exponentially hard one-way functions. 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 II, volume 4052 of Lecture Notes in Computer Science, pages 228-239. Springer, 2006. URL: http://dx.doi.org/10.1007/11787006_20.
http://dx.doi.org/10.1007/11787006_20
Iftach Haitner, Thomas Holenstein, Omer Reingold, Salil P. Vadhan, and Hoeteck Wee. Universal one-way hash functions via inaccessible entropy. In Henri Gilbert, editor, Advances in Cryptology - EUROCRYPT 2010, 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, French Riviera, May 30 - June 3, 2010. Proceedings, volume 6110 of Lecture Notes in Computer Science, pages 616-637. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13190-5_31.
http://dx.doi.org/10.1007/978-3-642-13190-5_31
Iftach Haitner, Minh-Huyen Nguyen, Shien Jin Ong, Omer Reingold, and Salil P. Vadhan. Statistically hiding commitments and statistical zero-knowledge arguments from any one-way function. SIAM J. Comput., 39(3):1153-1218, 2009. URL: http://dx.doi.org/10.1137/080725404.
http://dx.doi.org/10.1137/080725404
Iftach Haitner, Omer Reingold, and Salil P. Vadhan. Efficiency improvements in constructing pseudorandom generators from one-way functions. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 437-446. ACM, 2010. URL: http://dx.doi.org/10.1145/1806689.1806750.
http://dx.doi.org/10.1145/1806689.1806750
Iftach Haitner, Omer Reingold, and Salil P. Vadhan. Efficiency improvements in constructing pseudorandom generators from one-way functions. SIAM J. Comput., 42(3):1405-1430, 2013. URL: http://dx.doi.org/10.1137/100814421.
http://dx.doi.org/10.1137/100814421
Iftach Haitner, Omer Reingold, Salil P. Vadhan, and Hoeteck Wee. Inaccessible entropy. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 611-620. ACM, 2009. URL: http://dx.doi.org/10.1145/1536414.1536497.
http://dx.doi.org/10.1145/1536414.1536497
Johan Håstad, Russell Impagliazzo, Leonid A. Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. URL: http://dx.doi.org/10.1137/S0097539793244708.
http://dx.doi.org/10.1137/S0097539793244708
Thomas Holenstein. Pseudorandom generators from one-way functions: A simple construction for any hardness. In Shai Halevi and Tal Rabin, editors, Theory of Cryptography, Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006, Proceedings, volume 3876 of Lecture Notes in Computer Science, pages 443-461. Springer, 2006. URL: http://dx.doi.org/10.1007/11681878_23.
http://dx.doi.org/10.1007/11681878_23
Thomas Holenstein and Renato Renner. On the randomness of independent experiments. IEEE Trans. Information Theory, 57(4):1865-1871, 2011. URL: http://dx.doi.org/10.1109/TIT.2011.2110230.
http://dx.doi.org/10.1109/TIT.2011.2110230
Thomas Holenstein and Makrand Sinha. Constructing a pseudorandom generator requires an almost linear number of calls. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 698-707. IEEE, 2012.
Jonathan Katz and Chiu-Yuen Koo. On constructing universal one-way hash functions from arbitrary one-way functions. IACR Cryptology ePrint Archive, 2005:328, 2005. URL: http://eprint.iacr.org/2005/328.
http://eprint.iacr.org/2005/328
Shachar Lovett and Jiapeng Zhang. On the impossibility of entropy reversal, and its application to zero-knowledge proofs. In Theory of Cryptography Conference, pages 31-55. Springer, 2017.
Minh-Huyen Nguyen and Salil P. Vadhan. Zero knowledge with efficient provers. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 287-295. ACM, 2006. URL: http://dx.doi.org/10.1145/1132516.1132559.
http://dx.doi.org/10.1145/1132516.1132559
Tatsuaki Okamoto. On relationships between statistical zero-knowledge proofs. J. Comput. Syst. Sci., 60(1):47-108, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1664.
http://dx.doi.org/10.1006/jcss.1999.1664
Shien Jin Ong and Salil P. Vadhan. An equivalence between zero knowledge and commitments. In Ran Canetti, editor, Theory of Cryptography, Fifth Theory of Cryptography Conference, TCC 2008, New York, USA, March 19-21, 2008., volume 4948 of Lecture Notes in Computer Science, pages 482-500. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-78524-8_27.
http://dx.doi.org/10.1007/978-3-540-78524-8_27
Renato Renner and Stefan Wolf. Smooth rényi entropy and applications. In Information Theory, 2004. ISIT 2004. Proceedings. International Symposium on, page 233. IEEE, 2004.
John Rompel. One-way functions are necessary and sufficient for secure signatures. In Harriet Ortiz, editor, Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 387-394. ACM, 1990. URL: http://dx.doi.org/10.1145/100216.100269.
http://dx.doi.org/10.1145/100216.100269
Amit Sahai and Salil P. Vadhan. A complete promise problem for statistical zero-knowledge. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 448-457. IEEE Computer Society, 1997. URL: http://dx.doi.org/10.1109/SFCS.1997.646133.
http://dx.doi.org/10.1109/SFCS.1997.646133
Salil P. Vadhan and Colin Jia Zheng. Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 817-836. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214051.
http://dx.doi.org/10.1145/2213977.2214051
Salil Pravin Vadhan. A study of statistical zero-knowledge proofs. PhD thesis, Citeseer, 1999.
Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, and Jiapeng Zhang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Worst-Case to Average Case Reductions for the Distance to a Code
Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions vec{u}=(u_1,..., u_k), given as oracles, from a linear error correcting code V. The soundness of such systems relies on methods that act "locally" on vec{u} and map it to a single function u^* that is, roughly, as far from V as are u_1,..., u_k.
Motivated by these applications to efficient proof systems, we study a natural worst-case to average-case reduction of distance for linear spaces, and show several general cases in which the following statement holds: If some member of a linear space U=span(u_1,...,u_k) is delta-far from (all elements) of V in relative Hamming distance, then nearly all elements of U are (1-epsilon)delta-far from V; the value of epsilon depends only on the distance of the code V and approaches 0 as that distance approaches 1. Our results improve on the previous state-of-the-art which showed that nearly all elements of U are 1/2delta-far from V [Rothblum, Vadhan and Wigderson, STOC 2013].
When V is a Reed-Solomon (RS) code, as is often the case for algebraic proof systems, we show how to boost distance via a new "local" transformation that may be useful elsewhere. Relying on the affine-invariance of V, we map a vector u to a random linear combination of affine transformations of u, and show this process amplifies distance from V. Assuming V is an RS code with sufficiently large distance, this amplification process converts a function u that is somewhat far from V to one that is (1-epsilon)-far from V; as above, epsilon depends only on the distance of V and approaches 0 as the distance of V approaches 1.
We give two concrete application of these techniques. First, we revisit the axis-parallel low-degree test for bivariate polynomials of [Polischuk-Spielman, STOC 1994] and prove a "list-decoding" type result for it, when the degree of one axis is extremely small. This result is similar to the recent list-decoding-regime result of [Chiesa, Manohar and Shinkar, RANDOM 2017] but is proved using different techniques, and allows the degree in one axis to be arbitrarily large. Second, we improve the soundness analysis of the recent RS proximity testing protocol of [Ben-Sasson et al., ICALP 2018] and extend it to the "list-decoding" regime, bringing it closer to the Johnson bound.
Proximity testing
Reed-Solomon codes
algebraic coding complexity
Theory of computation~Error-correcting codes
24:1-24:23
Regular Paper
Work supported by the USA-Israel binational science fund, grant # 2014359
Eli
Ben-Sasson
Eli Ben-Sasson
Technion, Haifa, Israel
https://orcid.org/0000-0002-0708-0483
Supported by the European Research Council under POC grant OMIP -- DLV-693423 and Israel Science Foundation grant 1501/14.
Swastik
Kopparty
Swastik Kopparty
Rutgers Univeristy, New Brunswick, NJ, USA
Research supported in part by NSF grants CCF-1253886 and CCF-1540634.
Shubhangi
Saraf
Shubhangi Saraf
Rutgers Univeristy, New Brunswick, NJ, USA
Research supported in part by NSF grants CCF-1350572 and CCF-1540634.
10.4230/LIPIcs.CCC.2018.24
Scott Ames, Carmit Hazay, Yuval Ishai, and Muthuramakrishnan Venkitasubramaniam. Ligero: Lightweight sublinear arguments without a trusted setup. In Proceedings of the 24th ACM Conference on Computer and Communications Security, October 2017.
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, 1998. Preliminary version in FOCS '92.
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM, 45(1):70-122, 1998. Preliminary version in FOCS '92.
Sanjeev Arora and Madhu Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365-426, 2003. Preliminary version appeared in STOC '97.
László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, STOC '91, pages 21-32, 1991.
László Babai, Lance Fortnow, and Carsten Lund. Nondeterministic exponential time has two-prover interactive protocols. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, SFCS '90, pages 16-25, 1990.
László Babai and Shlomo Moran. Arthur-merlin games: A randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci., 36(2):254-276, 1988. URL: http://dx.doi.org/10.1016/0022-0000(88)90028-1.
http://dx.doi.org/10.1016/0022-0000(88)90028-1
Eli Ben-Sasson, Iddo Bentov, Yinon Horesh, and Michael Riabzev. Scalable, transparent, and post-quantum secure computational integrity. Cryptology ePrint Archive, Report 2018/046, 2018. Available at URL: https://eprint.iacr.org/2018/046.
https://eprint.iacr.org/2018/046
Eli Ben-Sasson, Iddo Bentov, Ynon Horesh, and Michael Riabzev. Fast Reed-Solomon Interactive Oracle Proofs of Proximity. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP), 2018. URL: https://eccc.weizmann.ac.il/report/2017/134.
https://eccc.weizmann.ac.il/report/2017/134
Eli Ben-Sasson, Alessandro Chiesa, Michael A. Forbes, Ariel Gabizon, Michael Riabzev, and Nicholas Spooner. On probabilistic checking in perfect zero knowledge. Electronic Colloquium on Computational Complexity (ECCC), 23:156, 2016. URL: http://eccc.hpi-web.de/report/2016/156.
http://eccc.hpi-web.de/report/2016/156
Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, and Madars Virza. Quasilinear-size zero knowledge from linear-algebraic PCPs. In Proceedings of the 13th Theory of Cryptography Conference, TCC '16, pages 33-64, 2016.
Eli Ben-Sasson, Alessandro Chiesa, and Nicholas Spooner. Interactive oracle proofs. In Martin Hirt and Adam D. Smith, editors, Theory of Cryptography - 14th International Conference, TCC 2016-B, Beijing, China, October 31 - November 3, 2016, Proceedings, Part II, volume 9986 of Lecture Notes in Computer Science, pages 31-60, 2016. URL: http://dx.doi.org/10.1007/978-3-662-53644-5_2.
http://dx.doi.org/10.1007/978-3-662-53644-5_2
Alessandro Chiesa, Peter Manohar, and Igor Shinkar. On axis-parallel tests for tensor product codes. In Klaus Jansen, José D. P. Rolim, David Williamson, and Santosh Srinivas Vempala, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, volume 81 of LIPIcs, pages 39:1-39:22. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.39.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.39
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 18(1):186-208, 1989. Preliminary version appeared in STOC '85.
Venkatesan Guruswami. Algorithmic results in list decoding. Foundations and Trends in Theoretical Computer Science, 2(2), 2006. URL: http://dx.doi.org/10.1561/0400000007.
http://dx.doi.org/10.1561/0400000007
Prahladh Harsha and Madhu Sudan. Small PCPs with low query complexity. Computational Complexity, 9(3-4):157-201, Dec 2000. Preliminary version in STACS '01.
Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992.
Alexander Polishchuk and Daniel A. Spielman. Nearly-linear size holographic proofs. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, STOC '94, pages 194-203, 1994.
Guy N. Rothblum, Salil Vadhan, and Avi Wigderson. Interactive proofs of proximity: delegating computation in sublinear time. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 793-802. ACM, 2013.
Eli Ben-Sasson, Swastik Kopparty, and Shubhangi Saraf
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Complexity of the Cayley Semigroup Membership Problem
We investigate the complexity of deciding, given a multiplication table representing a semigroup S, a subset X of S and an element t of S, whether t can be expressed as a product of elements of X. It is well-known that this problem is {NL}-complete and that the more general Cayley groupoid membership problem, where the multiplication table is not required to be associative, is {P}-complete. For groups, the problem can be solved in deterministic log-space which raised the question of determining the exact complexity of this variant. Barrington, Kadau, Lange and McKenzie showed that for Abelian groups and for certain solvable groups, the problem is contained in the complexity class {FOLL} and they concluded that these variants are not hard for any complexity class containing {Parity}. The more general case of arbitrary groups remained open. In this work, we show that for both groups and for commutative semigroups, the problem is solvable in {qAC}^0 (quasi-polynomial size circuits of constant depth with unbounded fan-in) and conclude that these variants are also not hard for any class containing {Parity}. Moreover, we prove that {NL}-completeness already holds for the classes of 0-simple semigroups and nilpotent semigroups. Together with our results on groups and commutative semigroups, we prove the existence of a natural class of finite semigroups which generates a variety of finite semigroups with {NL}-complete Cayley semigroup membership, while the Cayley semigroup membership problem for the class itself is not {NL}-hard. We also discuss applications of our technique to {FOLL}.
subsemigroup
multiplication table
generators
completeness
quasi-polynomial-size circuits
FOLL
Theory of computation~Problems, reductions and completeness
Theory of computation~Circuit complexity
25:1-25:12
Regular Paper
Lukas
Fleischer
Lukas Fleischer
FMI, University of Stuttgart , Universitätsstraße 38, 70569 Stuttgart, Germany
This work was supported by the DFG grant DI 435/5--2.
10.4230/LIPIcs.CCC.2018.25
Jorge Almeida. Some pseudovariety joins involving the pseudovariety of finite groups. Semigroup Forum, 37(1):53-57, Dec 1988. URL: http://dx.doi.org/10.1007/BF02573123.
http://dx.doi.org/10.1007/BF02573123
László Babai. Trading group theory for randomness. In Robert Sedgewick, editor, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, May 6-8, 1985, Providence, Rhode Island, USA, pages 421-429. ACM, 1985. URL: http://dx.doi.org/10.1145/22145.22192.
http://dx.doi.org/10.1145/22145.22192
László Babai. Local expansion of vertex-transitive graphs and random generation in finite groups. In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 164-174. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103440.
http://dx.doi.org/10.1145/103418.103440
László Babai, Robert Beals, Jin-Yi Cai, Gábor Ivanyos, and Eugene M. Luks. Multiplicative equations over commuting matrices. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '96, pages 498-507, Philadelphia, PA, USA, 1996. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=313852.314109.
http://dl.acm.org/citation.cfm?id=313852.314109
László Babai, Eugene M. Luks, and Ákos Seress. Permutation groups in NC. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 409-420. ACM, 1987. URL: http://dx.doi.org/10.1145/28395.28439.
http://dx.doi.org/10.1145/28395.28439
László Babai, Eugene M. Luks, and Ákos Seress. Permutation groups in NC. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 409-420. ACM, 1987. URL: http://dx.doi.org/10.1145/28395.28439.
http://dx.doi.org/10.1145/28395.28439
László Babai and Endre Szemerédi. On the complexity of matrix group problems I. In 25th Annual Symposium on Foundations of Computer Science, West Palm Beach, Florida, USA, 24-26 October 1984, pages 229-240. IEEE Computer Society, 1984. URL: http://dx.doi.org/10.1109/SFCS.1984.715919.
http://dx.doi.org/10.1109/SFCS.1984.715919
David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in nc¹. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 1-5. ACM, 1986. URL: http://dx.doi.org/10.1145/12130.12131.
http://dx.doi.org/10.1145/12130.12131
David A. Mix Barrington, Peter Kadau, Klaus-Jörn Lange, and Pierre McKenzie. On the complexity of some problems on groups input as multiplication tables. J. Comput. Syst. Sci., 63(2):186-200, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1764.
http://dx.doi.org/10.1006/jcss.2001.1764
David A. Mix Barrington and Pierre McKenzie. Oracle branching programs and logspace versus P. Inf. Comput., 95(1):96-115, 1991. URL: http://dx.doi.org/10.1016/0890-5401(91)90017-V.
http://dx.doi.org/10.1016/0890-5401(91)90017-V
David A. Mix Barrington and Denis Thérien. Finite monoids and the fine structure of NC¹. J. ACM, 35:941-952, 1988.
Martin Beaudry. Membership testing in commutative transformation semigroups. Inf. Comput., 79(1):84-93, 1988. URL: http://dx.doi.org/10.1016/0890-5401(88)90018-1.
http://dx.doi.org/10.1016/0890-5401(88)90018-1
Martin Beaudry. Membership Testing in Transformation Monoids. PhD thesis, McGill University, Montreal, Quebec, 1988.
Martin Beaudry. Membership testing in threshold one transformation monoids. Inf. Comput., 113(1):1-25, 1994. URL: http://dx.doi.org/10.1006/inco.1994.1062.
http://dx.doi.org/10.1006/inco.1994.1062
Martin Beaudry, Pierre McKenzie, and Denis Thérien. The membership problem in aperiodic transformation monoids. J. ACM, 39(3):599-616, 1992. URL: http://dx.doi.org/10.1145/146637.146661.
http://dx.doi.org/10.1145/146637.146661
Ravi B. Boppana. The average sensitivity of bounded-depth circuits. Inf. Process. Lett., 63(5):257-261, 1997. URL: http://dx.doi.org/10.1016/S0020-0190(97)00131-2.
http://dx.doi.org/10.1016/S0020-0190(97)00131-2
Xi Chen, Igor Carboni Oliveira, Rocco A. Servedio, and Li-Yang Tan. Near-optimal small-depth lower bounds for small distance connectivity. 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 612-625. ACM, 2016. URL: http://dx.doi.org/10.1145/2897518.2897534.
http://dx.doi.org/10.1145/2897518.2897534
Merrick L. Furst, John E. Hopcroft, and Eugene M. Luks. Polynomial-time algorithms for permutation groups. In 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, pages 36-41. IEEE Computer Society, 1980. URL: http://dx.doi.org/10.1109/SFCS.1980.34.
http://dx.doi.org/10.1109/SFCS.1980.34
Johan Håstad. Almost optimal lower bounds for small depth circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL: http://dx.doi.org/10.1145/12130.12132.
http://dx.doi.org/10.1145/12130.12132
Neil D. Jones and William T. Laaser. Complete problems for deterministic polynomial time. Theor. Comput. Sci., 3(1):105-117, 1976. URL: http://dx.doi.org/10.1016/0304-3975(76)90068-2.
http://dx.doi.org/10.1016/0304-3975(76)90068-2
Neil D. Jones, Y. Edmund Lien, and William T. Laaser. New problems complete for nondeterministic loc space. Mathematical Systems Theory, 10:1-17, 1976. URL: http://dx.doi.org/10.1007/BF01683259.
http://dx.doi.org/10.1007/BF01683259
P. Levi. Über die Untergruppen der freien Gruppen. (2. Mitteilung). Mathematische Zeitschrift, 37:90-97, 1933. URL: http://eudml.org/doc/168437.
http://eudml.org/doc/168437
Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4):17:1-17:24, 2008. URL: http://dx.doi.org/10.1145/1391289.1391291.
http://dx.doi.org/10.1145/1391289.1391291
Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles (preliminary version). In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21-23 October 1985, pages 1-10. IEEE Computer Society, 1985. URL: http://dx.doi.org/10.1109/SFCS.1985.49.
http://dx.doi.org/10.1109/SFCS.1985.49
Lukas Fleischer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth
We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.
Boolean circuits
semi-disjoint bilinear form
Boolean vector convolution
Boolean matrix product
Theory of computation~Circuit complexity
26:1-26:10
Regular Paper
Andrzej
Lingas
Andrzej Lingas
Department of Computer Science, Lund University, Box 118, 22100 Lund, Sweden
Research supported in part by VR grant 2017-03750.
10.4230/LIPIcs.CCC.2018.26
N. Alon and R. B. Boppana. The monotone circuit complexity of boolean functions. Combinatorica, 7(1):1-22, 1987.
A. E. Andreev. On one method of obtaining constructive lower bounds for the monotone circuit size. Algebra and Logics, 26(1):3-26, 1987.
N. Blum. An ω (n^4/3) lower bound on the monotone network complexity of the n-th degree convolution. Theoretical Computer Science, 36:59-69, 1985.
N. Blum. On negations in boolean networks. In Efficient Algorithms, volume 5760 of Lecture Notes in Computer Science, pages 18-29. Springer-Verlag, 2009.
J. H. Reif (editor). Synthesis of Parallel Algorithms. Morgan Kaufmann Publishers, San Mateo, 1993.
M. J. Fisher and M. S. Paterson. String-matching and other products. In Proceedings of the 7th SIAM-AMS Complexity of Computation, pages 113-125, 1974.
F. Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, Lecture Notes in Computer Science, pages 296-303. Springer-Verlag, 2014.
M. I. Grinchuk and I. S. Sergeev. Thin circulant matrices and lower bounds on the complexity of some boolean operations. Diskretn. Anal. Issled. Oper., 18:35-53, 2011.
K. Iwama and H. Morizumi. An explicit lower bound of 5n - o(n) for boolean circuits. In Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, pages 353-364. Springer-Verlag, 2002.
O. Lachish and R. Raz. Explicit lower bound of 4.5n - o(n) for boolen circuits. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 399-408. ACM, 2001.
E. A. Lamagna. The complexity of monotone networks for certain bilinear forms, routing problems, sorting, and merging. IEEE Transactions on Computers, c-28(10), 1979.
A. Lingas. Towards an almost quadratic lower bound on the monotone circuit complexity of the boolean convolution. In Theory and Applications of Models of Computation, Lecture Notes in Computer Science, pages 401-411. Springer-Verlag, 2017.
K. Mehlhorn and Z. Galil. Monotone switching circuits and boolean matrix product. Computing, 16:99-111, 1976.
M. Paterson. Complexity of monotone networks for boolean matrix product. Theoretical Computer Science, 1(1):13-20, 1975.
N. Pippenger and L.G. Valiant. Shifting graphs and their applications. Journal of the ACM, 23(3):423-432, 1976.
R. Pratt. The power of negative thinking in multiplying boolean matrices. SIAM J. Comput., 4(3):326-330, 1975.
R. Raz. On the complexity of matrix product. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 144-151. ACM, 2002.
A. A. Razborov. Lower bounds on the monotone complexity of some boolean functions. Doklady Akademii Nauk, 281(4):798-801, 1985.
C. P. Schnorr. Zwei lineare untere schranken für die komplexität boolescher funktionen. Computing, 13(2):155-171, 1974.
C. P. Schnorr. A lower bound on the number of additions in monotone computations. Theoretical Computer Science, 2(3):305-315, 1976.
A. Schönhage and V. Strassen. Schnelle multiplikation grober zahlen. Computing, 7:281-292, 1971.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354-356, 1969.
L.G. Valiant. Negation can be exponentially powerfull. Theoretical Computer Science, 12:303-314, 1980.
I. Wegener. The Complexity of Boolean Functions. Wiley-Teubner Series in Computer Science, New York, Stuggart, 1987.
J. Weiss. An n^3/2 lower bound on the monotone network complexity of the boolean convolution. Information and Control, 59:184-188, 1983.
V. Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 807-898. ACM, 2012.
U. Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM, 49(3):289-317, 2002.
Andrzej Lingas
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds on Non-Adaptive Data Structures Maintaining Sets of Numbers, from Sunflowers
We prove new cell-probe lower bounds for dynamic data structures that maintain a subset of {1,2,...,n}, and compute various statistics of the set. The data structure is said to handle insertions non-adaptively if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. For any such data structure that can compute the median of the set, we prove that: t_{med} >= Omega(n^{1/(t_{ins}+1)}/(w^2 * t_{ins}^2)), where t_{ins} is the number of memory locations accessed during insertions, t_{med} is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. When the data structure is able to perform deletions non-adaptively and compute the minimum non-adaptively, we prove t_{min} + t_{del} >= Omega(log n /(log w + log log n)), where t_{min} is the number of locations accessed to compute the minimum, and t_{del} is the number of locations accessed to perform deletions. For the predecessor search problem, where the data structure is required to compute the predecessor of any element in the set, we prove that if computing the predecessors can be done non-adaptively, then either t_{pred} >= Omega(log n/(log log n + log w)), or t_{ins} >= Omega(n^{1/(2(t_{pred}+1))}), where t_{pred} is the number of locations accessed to compute predecessors.
These bounds are nearly matched by Binary Search Trees in some range of parameters. Our results follow from using the Sunflower Lemma of Erdös and Rado [Paul Erdös and Richard Rado, 1960] together with several kinds of encoding arguments.
Non-adaptive data structures
Sunflower lemma
Theory of computation~Cell probe models and lower bounds
27:1-27:16
Regular Paper
Sivaramakrishnan
Natarajan Ramamoorthy
Sivaramakrishnan Natarajan Ramamoorthy
Paul G. Allen School for Computer Science & Engineering, University of Washington, Seattle, USA
Supported by the National Science Foundation under agreement CCF-1420268 and CCF-1524251
Anup
Rao
Anup Rao
Paul G. Allen School for Computer Science & Engineering, University of Washington, Seattle, USA
Supported by the National Science Foundation under agreement CCF-1420268 and CCF-1524251
10.4230/LIPIcs.CCC.2018.27
Miklós Ajtai. A lower bound for finding predecessors in yao’s call probe model. Combinatorica, 8(3), 1988.
Noga Alon and Ravi B. Boppana. The monotone circuit complexity of boolean functions. Combinatorica, 7(1):1-22, 1987.
Noga Alon and Uriel Feige. On the power of two, three and four probes. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009. SIAM, 2009.
Paul Beame and Faith E. Fich. Optimal bounds for the predecessor problem and related problems. JCSS: Journal of Computer and System Sciences, 65, 2002.
Paul Beame, Vincent Liew, and Mihai Patrascu. Finding the median (obliviously) with bounded space. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 103-115, 2015.
Joseph Boninger, Joshua Brody, and Owen Kephart. Non-adaptive data structure bounds for dynamic predecessor. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India, pages 20:1-20:12, 2017.
Gerth Stølting Brodal, Shiva Chaudhuri, and Jaikumar Radhakrishnan. The randomized complexity of maintaining the minimum. In SWAT: Scandinavian Workshop on Algorithm Theory, 1996.
Joshua Brody and Kasper Green Larsen. Adapt or die: Polynomial lower bounds for non-adaptive dynamic data structures. Theory of Computing, 11:471-489, 2015.
Amit Chakrabarti, T. S. Jayram, and Mihai Patrascu. Tight lower bounds for selection in randomly ordered streams. In Proc. 19th Symp. on Discrete Algorithms (SODA), pages 720-729. ACM/SIAM, 2008.
Timothy M. Chan. Comparison-based time-space lower bounds for selection. ACM Trans. Algorithms, 6(2), 2010.
Paul Erdős and Richard Rado. Intersection theorems for systems of sets. Journal of London Mathematical Society, 35:85-90, 1960.
Gudmund Skovbjerg Frandsen, Peter Bro Miltersen, and Sven Skyum. Dynamic word problems. J. ACM, 44(2):257-271, 1997. URL: http://dx.doi.org/10.1145/256303.256309.
http://dx.doi.org/10.1145/256303.256309
Michael Fredman and Michael Saks. The cell probe complexity of dynamic data structures. In STOC: ACM Symposium on Theory of Computing (STOC), 1989.
Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. JCSS: Journal of Computer and System Sciences, 47, 1993.
Anna Gál and Peter Bro Miltersen. The cell probe complexity of succinct data structures. Theor. Comput. Sci., 379(3):405-417, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.02.047.
http://dx.doi.org/10.1016/j.tcs.2007.02.047
Mohit Garg and Jaikumar Radhakrishnan. Set membership with non-adaptive bit probes. In 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, pages 38:1-38:13, 2017.
Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In STOC: ACM Symposium on Theory of Computing (STOC), 2000.
Kasper Green Larsen. The cell probe complexity of dynamic range counting. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 85-94, 2012.
Peter Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. 57:37-49, 1 1998.
Peter Bro Miltersen. Lower bounds for union-split-find related problems on random access machines. In Proceedings of the 26th Annual Symposium on the Theory of Computing, pages 625-634, New York, 1994. ACM Press.
J. Ian Munro and Venkatesh Raman. Selection from read-only memory and sorting with minimum data movement. TCS: Theoretical Computer Science, 165, 1996.
Rina Panigrahy, Kunal Talwar, and Udi Wieder. Lower bounds on near neighbor search via metric expansion. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 805-814, Washington, DC, USA, 2010. IEEE Computer Society.
Mihai Pǎtraşcu. Lower bounds for 2-dimensional range counting. In Proc. 39th ACM Symposium on Theory of Computing (STOC), pages 40-46, 2007.
Mihai Pǎtraşcu and Erik D. Demaine. Logarithmic lower bounds in the cell-probe model. SIAM Journal on Computing, 35(4):932-963, 2006. See also STOC'04, SODA'04.
Mihai Patrascu and Mikkel Thorup. Time-space trade-offs for predecessor search. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 232-240, 2006.
Mihai Pǎtraşcu and Mikkel Thorup. Don't rush into a union: Take time to find your roots. In Proc. 43rd ACM Symposium on Theory of Computing (STOC), pages 559-568, 2011. See also arXiv:1102.1783.
Mihai Patrascu and Mikkel Thorup. Dynamic integer sets with optimal rank, select, and predecessor search. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 166-175, 2014.
Pranab Sen and Srinivasan Venkatesh. Lower bounds for predecessor searching in the cell probe model. J. Comput. Syst. Sci, 74(3):364-385, 2008.
Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Information Processing Letters, 6(3):80-82, 1977.
Omri Weinstein and Huacheng Yu. Amortized dynamic cell-probe lower bounds from four-party communication. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 305-314, 2016.
Dan E. Willard. Log-logarithmic worst-case range queries are possible in space Θ(N). Information Processing Letters, pages 81-84, 1983.
Andrew Yao. Should tables be sorted? JACM: Journal of the ACM, 28, 1981.
Huacheng Yu. Cell-probe lower bounds for dynamic problems via a new communication model. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 362-374, 2016.
Sivaramakrishnan Natarajan Ramamoorthy and Anup Rao
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Dimension Reduction for Polynomials over Gaussian Space and Applications
We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.
Dimension reduction
Low-degree Polynomials
Noise Stability
Non-Interactive Simulation
Theory of computation~Complexity theory and logic
28:1-28:37
Regular Paper
Badih
Ghazi
Badih Ghazi
Google Research, 1600 Amphitheatre Parkway Mountain View, CA 94043, USA
The work was done while the author was a student at MIT. Supported in parts by NSF CCF-1650733 and CCF-1420692.
Pritish
Kamath
Pritish Kamath
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Supported in parts by NSF CCF-1420956, CCF-1420692, CCF-1218547 and CCF-1650733.
Prasad
Raghavendra
Prasad Raghavendra
University of California Berkeley, Berkeley, CA, USA
Research supported by Okawa Research Grant and NSF CCF-1408643.
10.4230/LIPIcs.CCC.2018.28
OpenQIProblemsWiki - All the Bell Inequalities. http://qig.itp.uni-hannover.de/qiproblems/1. Accessed: 2016-07-12.
http://qig.itp.uni-hannover.de/qiproblems/1
Rudolf Ahlswede and Imre Csiszár. Common randomness in information theory and cryptography. part i: secret sharing. IEEE Transactions on Information Theory, 39(4), 1993.
Rudolf Ahlswede and Imre Csiszár. Common randomness in information theory and cryptography. ii. cr capacity. Information Theory, IEEE Transactions on, 44(1):225-240, 1998.
Noga Alon and Eyal Lubetzky. The shannon capacity of a graph and the independence numbers of its powers. Information Theory, IEEE Transactions on, 52(5):2172-2176, 2006.
Boaz Barak, Moritz Hardt, Ishay Haviv, Anup Rao, Oded Regev, and David Steurer. Rounding parallel repetitions of unique games. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 374-383. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.55.
http://dx.doi.org/10.1109/FOCS.2008.55
Mohammad Bavarian, Dmitry Gavinsky, and Tsuyoshi Ito. On the role of shared randomness in simultaneous communication. In Automata, Languages, and Programming, pages 150-162. Springer, 2014.
Salman Beigi. A new quantum data processing inequality. CoRR, abs/1210.1689, 2012. URL: http://arxiv.org/abs/1210.1689.
http://arxiv.org/abs/1210.1689
Salman Beigi and Amin Gohari. On the duality of additivity and tensorization. arXiv preprint arXiv:1502.00827, 2015.
Andrej Bogdanov and Elchanan Mossel. On extracting common random bits from correlated sources. Information Theory, IEEE Transactions on, 57(10):6351-6355, 2011.
Christer Borell. Geometric bounds on the ornstein-uhlenbeck velocity process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 70(1):1-13, 1985.
Gilles Brassard and Louis Salvail. Secret-key reconciliation by public discussion. In advances in Cryptology—EUROCRYPT’93, pages 410-423. Springer, 1994.
Mark Braverman and Young Kun-Ko. Information value of two-prover games. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 12:1-12:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.12.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.12
Mark Braverman and Anup Rao. Information equals amortized communication. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 748-757. IEEE, 2011.
Mark Braverman and Jon Schneider. Information complexity is computable. arXiv preprint arXiv:1502.02971, 2015.
Clement Cannonne, Venkat Guruswami, Raghu Meka, and Madhu Sudan. Communication with imperfectly shared randomness. ITCS, 2014.
Eric Chitambar, Runyao Duan, and Yaoyun Shi. Tripartite entanglement transformations and tensor rank. Physical review letters, 101(14):140502, 2008.
Imre Csiszár and Prakash Narayan. Common randomness and secret key generation with a helper. Information Theory, IEEE Transactions on, 46(2):344-366, 2000.
Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of johnson and lindenstrauss. Random Struct. Algorithms, 22(1):60-65, 2003. URL: http://dx.doi.org/10.1002/rsa.10073.
http://dx.doi.org/10.1002/rsa.10073
Anindya De, Elchanan Mossel, and Joe Neeman. Noise stability is computable and approximately low-dimensional. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 10:1-10:11. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.10.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.10
Anindya De, Elchanan Mossel, and Joe Neeman. Non interactive simulation of correlated distributions is decidable. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2728-2746. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.174.
http://dx.doi.org/10.1137/1.9781611975031.174
Anindya De and Rocco A Servedio. Efficient deterministic approximate counting for low-degree polynomial threshold functions. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 832-841. ACM, 2014.
Payam Delgosha and Salman Beigi. Impossibility of local state transformation via hypercontractivity. CoRR, abs/1307.2747, 2013. URL: http://arxiv.org/abs/1307.2747.
http://arxiv.org/abs/1307.2747
Payam Delgosha and Salman Beigi. Impossibility of local state transformation via hypercontractivity. Communications in Mathematical Physics, 332(1):449-476, 2014.
Peter Gács and János Körner. Common information is far less than mutual information. Problems of Control and Information Theory, 2(2):149-162, 1973.
Hans Gebelein. Das statistische problem der korrelation als variations-und eigenwertproblem und sein zusammenhang mit der ausgleichsrechnung. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 21(6):364-379, 1941.
Badih Ghazi and T. S. Jayram. Resource-efficient common randomness and secret-key schemes. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1834-1853. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.120.
http://dx.doi.org/10.1137/1.9781611975031.120
Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension reduction for polynomials over gaussian space and applications. Electronic Colloquium on Computational Complexity (ECCC), 24:125, 2017. URL: https://eccc.weizmann.ac.il/report/2017/125.
https://eccc.weizmann.ac.il/report/2017/125
Badih Ghazi, Pritish Kamath, and Madhu Sudan. Communication complexity of permutation-invariant functions. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1902-1921. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch134.
http://dx.doi.org/10.1137/1.9781611974331.ch134
Badih Ghazi, Pritish Kamath, and Madhu Sudan. Decidability of non-interactive simulation of joint distributions. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 545-554. IEEE Computer Society, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.65.
http://dx.doi.org/10.1109/FOCS.2016.65
Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL: http://dx.doi.org/10.1145/227683.227684.
http://dx.doi.org/10.1145/227683.227684
Steven Heilman. Euclidean partitions optimizing noise stability. CoRR, abs/1211.7138, 2012. URL: http://arxiv.org/abs/1211.7138.
http://arxiv.org/abs/1211.7138
Steven Heilman, Elchanan Mossel, and Joe Neeman. Standard simplices and pluralities are not the most noise stable. Israel Journal of Mathematics, 213(1):33-53, 2016.
Hermann O. Hirschfeld. A connection between correlation and contingency. Mathematical Proceedings of the Cambridge Philosophical Society, 31(4):520–-524, 1935. URL: http://dx.doi.org/10.1017/S0305004100013517.
http://dx.doi.org/10.1017/S0305004100013517
Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(1):141-172, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a008.
http://dx.doi.org/10.4086/toc.2009.v005a008
Marcus Isaksson and Elchanan Mossel. Maximally stable gaussian partitions with discrete applications. Israel Journal of Mathematics, 189(1):347-396, 2012.
William Johnson and Joram Lindenstrauss. Extensions of lipschitz maps into a hilbert space. Contemporary Mathematics, 26:189-206, 01 1984.
Sudeep Kamath and Venkat Anantharam. On non-interactive simulation of joint distributions. IEEE Trans. Information Theory, 62(6):3419-3435, 2016. URL: http://dx.doi.org/10.1109/TIT.2016.2553672.
http://dx.doi.org/10.1109/TIT.2016.2553672
Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, and Thomas Vidick. Entangled games are hard to approximate. SIAM J. Comput., 40(3):848-877, 2011. URL: http://dx.doi.org/10.1137/090751293.
http://dx.doi.org/10.1137/090751293
Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37(1):319-357, 2007.
László Lovász. On the shannon capacity of a graph. Information Theory, IEEE Transactions on, 25(1):1-7, 1979.
Ueli M Maurer. Secret key agreement by public discussion from common information. Information Theory, IEEE Transactions on, 39(3):733-742, 1993.
Elchanan Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis, 19(6):1713-1756, 2010.
Elchanan Mossel and Ryan O'Donnell. Coin flipping from a cosmic source: On error correction of truly random bits. arXiv preprint math/0406504, 2004.
Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on, pages 21-30. IEEE, 2005.
Elchanan Mossel, Ryan O'Donnell, Oded Regev, Jeffrey E Steif, and Benny Sudakov. Non-interactive correlation distillation, inhomogeneous markov chains, and the reverse bonami-beckner inequality. Israel Journal of Mathematics, 154(1):299-336, 2006.
Ilan Newman. Private vs. common random bits in communication complexity. Information processing letters, 39(2):67-71, 1991.
Michael A Nielsen. Conditions for a class of entanglement transformations. Physical Review Letters, 83(2):436, 1999.
Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM J. Comput., 40(6):1871-1891, 2011. URL: http://dx.doi.org/10.1137/080734042.
http://dx.doi.org/10.1137/080734042
Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: http://dx.doi.org/10.1137/S0097539795280895.
http://dx.doi.org/10.1137/S0097539795280895
Ran Raz. A counterexample to strong parallel repetition. SIAM J. Comput., 40(3):771-777, 2011. URL: http://dx.doi.org/10.1137/090747270.
http://dx.doi.org/10.1137/090747270
Renato Renner and Stefan Wolf. Simple and tight bounds for information reconciliation and privacy amplification. In Advances in cryptology-ASIACRYPT 2005, pages 199-216. Springer, 2005.
Alfréd Rényi. On measures of dependence. Acta mathematica hungarica, 10(3-4):441-451, 1959.
Claude E Shannon. The zero error capacity of a noisy channel. Information Theory, IRE Transactions on, 2(3):8-19, 1956.
Hans S Witsenhausen. On sequences of pairs of dependent random variables. SIAM Journal on Applied Mathematics, 28(1):100-113, 1975.
Pawel Wolff. Hypercontractivity of simple random variables. Studia Mathematica, 180(3):219-236, 2007.
Aaron D. Wyner. The common information of two dependent random variables. IEEE Trans. Information Theory, 21(2):163-179, 1975. URL: http://dx.doi.org/10.1109/TIT.1975.1055346.
http://dx.doi.org/10.1109/TIT.1975.1055346
Badih Ghazi, Pritish Kamath, and Prasad Raghavendra
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode