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Lower Bounds for Set-Multilinear Branching Programs

Authors Prerona Chatterjee , Deepanshu Kush , Shubhangi Saraf , Amir Shpilka

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Author Details

Prerona Chatterjee
  • School of Computer Sciences, NISER Bhubaneswar, India
Deepanshu Kush
  • Department of Computer Science, University of Toronto, Canada
Shubhangi Saraf
  • Department of Computer Science, University of Toronto, Canada
  • Department of Mathematics, University of Toronto, Canada
Amir Shpilka
  • Blavatnik School of Computer Science, Tel-Aviv University, Israel

Funding

  • Chatterjee, Prerona: This work was done as a postdoctoral student at Tel Aviv University, where the research was funded by the Azrieli International Postdoctoral Fellowship, the Israel Science Foundation (grant number 514/20) and the Len Blavatnik and the Blavatnik Family foundation.
  • Saraf, Shubhangi: Research partially supported by a Sloan research fellowship and an NSERC Discovery Grant.
  • Shpilka, Amir: Research leading to these results has received funding from the Israel Science Foundation (grant number 514/20) and from the Len Blavatnik and the Blavatnik Family foundation.

Acknowledgements

Parts of this work were done while the first author was visiting TIFR Mumbai and ICTS-TIFR Bengaluru, and she would like to thank Venkata Susmita Biswas, Ramprasad Saptharishi, Prahladh Harsha and Jaikumar Radhakrishnan for the hospitality. The first author would also like to thank Anamay Tengse for useful discussions.

Cite AsGet BibTex

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.20

Abstract

In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Lower Bounds
  • Algebraic Branching Programs
  • Set-multilinear polynomials

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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. Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk. Identity testing and lower bounds for read-k oblivious algebraic branching programs. ACM Trans. Comput. Theory, 10(1):3:1-3:30, 2018. URL: https://doi.org/10.1145/3170709.
  3. Vikraman Arvind and S. Raja. Some lower bound results for set-multilinear arithmetic computations. Chic. J. Theor. Comput. Sci., 2016, 2016. URL: http://cjtcs.cs.uchicago.edu/articles/2016/6/contents.html.
  4. C. S. Bhargav, Sagnik Dutta, and Nitin Saxena. Improved lower bound, and proof barrier, for constant depth algebraic circuits. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022, August 22-26, 2022, Vienna, Austria, volume 241 of LIPIcs, pages 18:1-18:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.MFCS.2022.18.
  5. C.S. Bhargav, Prateek Dwivedi, and Nitin Saxena. Lower bounds for the sum of small-size algebraic branching programs. To appear in the proceedings of the Annual Conference on Theory and Applications of Models of Computation, 2024. URL: https://www.cse.iitk.ac.in/users/nitin/papers/sumRO.pdf.
  6. Vishwas Bhargava and Sumanta Ghosh. Improved hitting set for orbit of roabps. Comput. Complex., 31(2):15, 2022. URL: https://doi.org/10.1007/S00037-022-00230-9.
  7. Pranav Bisht and Nitin Saxena. Blackbox identity testing for sum of special roabps and its border class. Comput. Complex., 30(1):8, 2021. URL: https://doi.org/10.1007/S00037-021-00209-Y.
  8. Peter Bürgisser. Cook’s versus valiant’s hypothesis. Theor. Comput. Sci., 235(1):71-88, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00183-8.
  9. Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. Quadratic lower bounds for algebraic branching programs and formulas. Comput. Complex., 31(2):8, 2022. URL: https://doi.org/10.1007/S00037-022-00223-8.
  10. Zeev Dvir, Guillaume Malod, Sylvain Perifel, and Amir Yehudayoff. Separating multilinear branching programs and formulas. 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 615-624. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214034.
  11. Michael A. Forbes, Ramprasad Saptharishi, and Amir Shpilka. Hitting sets for multilinear read-once algebraic branching programs, in any order. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 867-875. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591816.
  12. 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.
  13. Purnata Ghosal and B. V. Raghavendra Rao. Limitations of sums of bounded read formulas and abps. In Rahul Santhanam and Daniil Musatov, editors, Computer Science - Theory and Applications - 16th International Computer Science Symposium in Russia, CSR 2021, Sochi, Russia, June 28 - July 2, 2021, Proceedings, volume 12730 of Lecture Notes in Computer Science, pages 147-169. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-79416-3_9.
  14. Zeyu Guo and Rohit Gurjar. Improved explicit hitting-sets for roabps. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  15. Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Identity testing for constant-width, and any-order, read-once oblivious arithmetic branching programs. Theory Comput., 13(1):1-21, 2017. URL: https://doi.org/10.4086/TOC.2017.V013A002.
  16. Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic identity testing for sum of read-once oblivious arithmetic branching programs. Comput. Complex., 26(4):835-880, 2017. URL: https://doi.org/10.1007/S00037-016-0141-Z.
  17. Rohit Gurjar and Ben Lee Volk. Pseudorandom bits for oblivious branching programs. ACM Trans. Comput. Theory, 12(2):8:1-8:12, 2020. URL: https://doi.org/10.1145/3378663.
  18. 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: https://doi.org/10.1007/978-3-540-85238-4_33.
  19. Neeraj Kayal. An exponential lower bound for the sum of powers of bounded degree polynomials. Electron. Colloquium Comput. Complex., TR12-081, 2012. URL: https://arxiv.org/abs/TR12-081.
  20. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An exponential lower bound for homogeneous depth four arithmetic formulas. SIAM J. Comput., 46(1):307-335, 2017. URL: https://doi.org/10.1137/151002423.
  21. 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. URL: https://doi.org/10.1145/3369928.
  22. 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: https://doi.org/10.1145/2591796.2591847.
  23. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. An almost cubic lower bound for depth three arithmetic circuits. 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 33:1-33:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.33.
  24. Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. SIAM J. Comput., 46(1):336-387, 2017. URL: https://doi.org/10.1137/140999335.
  25. Deepanshu Kush and Shubhangi Saraf. Improved low-depth set-multilinear circuit lower bounds. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 38:1-38:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.38.
  26. Deepanshu Kush and Shubhangi Saraf. Near-optimal set-multilinear formula lower bounds. In Amnon Ta-Shma, editor, 38th Computational Complexity Conference, CCC 2023, July 17-20, 2023, Warwick, UK, volume 264 of LIPIcs, pages 15:1-15:33. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.15.
  27. 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.
  28. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the partial derivative method applied to lopsided set-multilinear polynomials. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 32:1-32:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CCC.2022.32.
  29. 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: https://doi.org/10.1145/103418.103462.
  30. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex., 6(3):217-234, 1997. URL: https://doi.org/10.1007/BF01294256.
  31. C. Ramya and B. V. Raghavendra Rao. Lower bounds for special cases of syntactic multilinear abps. Theor. Comput. Sci., 809:1-20, 2020. URL: https://doi.org/10.1016/J.TCS.2019.10.047.
  32. Ran Raz. Separation of multilinear circuit and formula size. Theory Comput., 2(6):121-135, 2006. URL: https://doi.org/10.4086/toc.2006.v002a006.
  33. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. URL: https://doi.org/10.1145/2535928.
  34. Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Comput. Complex., 14(1):1-19, 2005. URL: https://doi.org/10.1007/S00037-005-0188-8.
  35. Ran Raz and Amir Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Comput. Complex., 17(4):515-535, 2008. URL: https://doi.org/10.1007/S00037-008-0254-0.
  36. Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Comput. Complex., 18(2):171-207, 2009. URL: https://doi.org/10.1007/s00037-009-0270-8.
  37. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github Survey, 2015. URL: https://github.com/dasarpmar/lowerbounds-survey.
  38. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5(3-4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  39. Sébastien Tavenas, Nutan Limaye, and Srikanth Srinivasan. Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplication. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 416-425. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520044.
  40. L. G. Valiant. Completeness classes in algebra. In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC '79, pages 249-261, New York, NY, USA, 1979. Association for Computing Machinery. URL: https://doi.org/10.1145/800135.804419.
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