NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors Omkar Baraskar, Agrim Dewan, Chandan Saha, Pulkit Sinha



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.16.pdf
  • Filesize: 0.84 MB
  • 21 pages

Document Identifiers

Author Details

Omkar Baraskar
  • University of Waterloo, Canada
Agrim Dewan
  • Indian Institute of Science, Bengaluru, India
Chandan Saha
  • Indian Institute of Science, Bengaluru, India
Pulkit Sinha
  • University of Waterloo, Canada

Acknowledgements

We thank the anonymous reviewers for their detailed and constructive feedback, which has helped us improve the presentation of this work. In particular, we thank one of the reviewers for pointing out some inaccuracies in the original proofs of Lemmas 46 and 47; simpler proofs for both lemmas came up in the process of fixing these inaccuracies.

Cite AsGet BibTex

Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.16

Abstract

An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Equivalence testing
  • MCSP
  • sparse polynomials
  • 3SAT

Metrics

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

References

  1. Manindra Agrawal, Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits. SIAM J. Comput., 44(3):669-697, 2015. URL: https://doi.org/10.1137/140975103.
  2. Manindra Agrawal and Nitin Saxena. Automorphisms of finite rings and applications to complexity of problems. In Proceedings of the 22nd Annual Conference on Theoretical Aspects of Computer Science, STACS'05, pages 1-17, Berlin, Heidelberg, 2005. Springer-Verlag. URL: https://doi.org/10.1007/978-3-540-31856-9_1.
  3. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. Inf. Comput., 256:2-8, 2017. Conference version appeared in the proceedings of MFCS 2014. URL: https://doi.org/10.1016/J.IC.2017.04.004.
  4. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing DNF Formulas and AC^0_d Circuits Given a Truth Table. In 21st Annual IEEE Conference on Computational Complexity (CCC 2006), 16-20 July 2006, Prague, Czech Republic, pages 237-251. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/CCC.2006.27.
  5. Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, and Bhargav Thankey. Low-depth arithmetic circuit lower bounds: Bypassing set-multilinearization. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 12:1-12:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.12.
  6. 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. Conference version appeared in the proceedings of FOCS 1992. URL: https://doi.org/10.1145/278298.278306.
  7. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. Conference version appeared in the proceedings of FOCS 1992. URL: https://doi.org/10.1145/273865.273901.
  8. Omkar Baraskar, Agrim Dewan, and Chandan Saha. Testing equivalence to design polynomials. In Olaf Beyersdorff, Mamadou Moustapha Kanté, Orna Kupferman, and Daniel Lokshtanov, editors, 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024, March 12-14, 2024, Clermont-Ferrand, France, volume 289 of LIPIcs, pages 9:1-9:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.STACS.2024.9.
  9. Amos Beimel, Francesco Bergadano, Nader H. Bshouty, Eyal Kushilevitz, and Stefano Varricchio. Learning functions represented as multiplicity automata. J. ACM, 47(3):506-530, 2000. Conference version appeared in the proceedings of FOCS 1996. URL: https://doi.org/10.1145/337244.337257.
  10. Michael Ben-Or and Prasoon Tiwari. A Deterministic Algorithm for Sparse Multivariate Polynominal Interpolation (Extended Abstract). In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 301-309. ACM, 1988. URL: https://doi.org/10.1145/62212.62241.
  11. Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. Deterministic factorization of sparse polynomials with bounded individual degree. J. ACM, 67(2):8:1-8:28, 2020. Conference version appeared in the proceedings of FOCS 2018. URL: https://doi.org/10.1145/3365667.
  12. Markus Bläser and Gorav Jindal. A new deterministic algorithm for sparse multivariate polynomial interpolation. In Katsusuke Nabeshima, Kosaku Nagasaka, Franz Winkler, and Ágnes Szántó, editors, International Symposium on Symbolic and Algebraic Computation, ISSAC '14, Kobe, Japan, July 23-25, 2014, pages 51-58. ACM, 2014. URL: https://doi.org/10.1145/2608628.2608648.
  13. Lenore Blum, Mike Shub, and Steve Smale. On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, Recursive Functions and Universal Machines. Bulletin of the American Mathematical Society, 21(1):1-46, 1989. URL: https://doi.org/10.1090/S0273-0979-1989-15750-9.
  14. 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 für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.CCC.2016.10.
  15. Suryajith Chillara, Coral Grichener, and Amir Shpilka. On hardness of testing equivalence to sparse polynomials under shifts. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 22:1-22:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.22.
  16. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL: https://doi.org/10.1016/0020-0190(78)90067-4.
  17. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. Conference version appeared in the proceedings of STOC 2006. URL: https://doi.org/10.1145/1236457.1236459.
  18. Michael A. Forbes. Some concrete questions on the border complexity of polynomials, 2016. URL: https://www.youtube.com/watch?v=1HMogQIHT6Q.
  19. Michael A. Forbes and Amir Shpilka. Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 243-252. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.34.
  20. Dima Grigoriev and Marek Karpinski. A Zero-Test and an Interpolation Algorithm for the Shifted Sparse Polynominals. In Gérard D. Cohen, Teo Mora, and Oscar Moreno, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 10th International Symposium, AAECC-10, San Juan de Puerto Rico, Puerto Rico, May 10-14, 1993, Proceedings, volume 673 of Lecture Notes in Computer Science, pages 162-169. Springer, 1993. URL: https://doi.org/10.1007/3-540-56686-4_41.
  21. Dima Grigoriev, Marek Karpinski, and Michael F. Singer. Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields. SIAM J. Comput., 19(6):1059-1063, 1990. URL: https://doi.org/10.1137/0219073.
  22. Dima Grigoriev and Alexander A. Razborov. Exponential Complexity Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions Over Finite Fields. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 269-278. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743456.
  23. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM J. Comput., 45(3):1064-1079, 2016. Conference version appeared in the proceedings of FOCS 2013. URL: https://doi.org/10.1137/140957123.
  24. Nikhil Gupta, Chandan Saha, and Bhargav Thankey. Equivalence Test for Read-Once Arithmetic Formulas. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 4205-4272. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch162.
  25. Thomas R. Hancock, Tao Jiang, Ming Li, and John Tromp. Lower Bounds on Learning Decision Lists and Trees. Inf. Comput., 126(2):114-122, 1996. URL: https://doi.org/10.1006/INCO.1996.0040.
  26. Johan Håstad. Tensor Rank is NP-Complete. J. Algorithms, 11(4):644-654, 1990. Conference version appeared in the proceedings of ICALP 1989. URL: https://doi.org/10.1016/0196-6774(90)90014-6.
  27. Shuichi Hirahara. Non-Black-Box Worst-Case to Average-Case Reductions within NP. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 247-258. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00032.
  28. Shuichi Hirahara. NP-Hardness of Learning Programs and Partial MCSP. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 968-979. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00095.
  29. Shuichi Hirahara, Igor C. Oliveira, and Rahul Santhanam. NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference, CCC 2018, June 22-24, 2018, San Diego, CA, USA, volume 102 of LIPIcs, pages 5:1-5:31. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.CCC.2018.5.
  30. Rahul Ilango. Constant Depth Formula and Partial Function Versions of MCSP are Hard. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 424-433. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00047.
  31. Rahul Ilango. The Minimum Formula Size Problem is (ETH) Hard. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 427-432. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00050.
  32. Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 22:1-22:36. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.CCC.2020.22.
  33. Russell Impagliazzo and Avi Wigderson. P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. 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 220-229. ACM, 1997. URL: https://doi.org/10.1145/258533.258590.
  34. 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: https://doi.org/10.1145/335305.335314.
  35. Erich L. Kaltofen and Barry M. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symb. Comput., 9(3):301-320, 1990. Conference version appeared in the proceedings of FOCS 1988. URL: https://doi.org/10.1016/S0747-7171(08)80015-6.
  36. Neeraj Kayal. Efficient algorithms for some special cases of the polynomial equivalence problem. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1409-1421. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.108.
  37. Neeraj Kayal. Affine projections of polynomials: extended abstract. 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 643-662. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214036.
  38. Neeraj Kayal, Vineet Nair, and Chandan Saha. Separation between read-once oblivious algebraic branching programs (roabps) and multilinear depth-three circuits. ACM Trans. Comput. Theory, 12(1):2:1-2:27, 2020. Conference version appeared in the proceedings of STACS 2016. URL: https://doi.org/10.1145/3369928.
  39. Neeraj Kayal, Vineet Nair, Chandan Saha, and Sébastien Tavenas. Reconstruction of full rank algebraic branching programs. ACM Trans. Comput. Theory, 11(1):2:1-2:56, 2019. Conference version appeared in the proceedings of CCC 2017. URL: https://doi.org/10.1145/3282427.
  40. Adam R. Klivans and Amir Shpilka. Learning restricted models of arithmetic circuits. Theory Comput., 2(10):185-206, 2006. Conference version appeared in the proceedings of COLT 2003. URL: https://doi.org/10.4086/TOC.2006.V002A010.
  41. Adam R. Klivans and Daniel A. Spielman. Randomness efficient identity testing of multivariate polynomials. 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 216-223. ACM, 2001. URL: https://doi.org/10.1145/380752.380801.
  42. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 804-814. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00083.
  43. Richard J. Lipton and Nisheeth K. Vishnoi. Deterministic identity testing for multivariate polynomials. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 756-760. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644233.
  44. W. J. Masek. Some NP-complete set covering problems. Unpublished Manuscript, 1979. Google Scholar
  45. Dori Medini and Amir Shpilka. Hitting sets and reconstruction for dense orbits in vp_eand ΣΠΣ circuits. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 19:1-19:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.CCC.2021.19.
  46. Jacques Patarin. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In Ueli M. Maurer, editor, Advances in Cryptology - EUROCRYPT '96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, volume 1070 of Lecture Notes in Computer Science, pages 33-48. Springer, 1996. URL: https://doi.org/10.1007/3-540-68339-9_4.
  47. Ján Pich and Rahul Santhanam. Why are Proof Complexity Lower Bounds Hard? In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1305-1324. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00080.
  48. Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Math. Notes, 41:333-338, 1987. URL: https://doi.org/10.1007/BF01137685.
  49. Daniel S. Roche. What Can (and Can't) we Do with Sparse Polynomials? In Manuel Kauers, Alexey Ovchinnikov, and Éric Schost, editors, Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2018, New York, NY, USA, July 16-19, 2018, pages 25-30. ACM, 2018. URL: https://doi.org/10.1145/3208976.3209027.
  50. Chandan Saha and Bhargav Thankey. Hitting sets for orbits of circuit classes and polynomial families. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 50:1-50:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2021.50.
  51. Nitin Saxena. Morphisms of rings and applications to complexity. PhD thesis, Indian Institute of Technology, Kanpur, 2006. URL: https://www.cse.iitk.ac.in/users/manindra/Students/thesis_saxena.pdf.
  52. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  53. Yaroslav Shitov. How hard is the tensor rank?, 2016. URL: https://arxiv.org/abs/1611.01559.
  54. Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Inf. Comput., 240:2-11, 2015. Conference version appeared in the proceedings of MFCS 2013. URL: https://doi.org/10.1016/J.IC.2014.09.004.
  55. Thomas Thierauf. The isomorphism problem for read-once branching programs and arithmetic circuits. Chic. J. Theor. Comput. Sci., 1998, 1998. URL: http://cjtcs.cs.uchicago.edu/articles/1998/1/contents.html.
  56. Joachim von zur Gathen and Erich L. Kaltofen. Factoring Sparse Multivariate Polynomials. J. Comput. Syst. Sci., 31(2):265-287, 1985. URL: https://doi.org/10.1016/0022-0000(85)90044-3.
  57. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation, EUROSAM '79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings, pages 216-226, 1979. URL: https://doi.org/10.1007/3-540-09519-5_73.