Tensor Ranks and the Fine-Grained Complexity of Dynamic Programming

Authors Josh Alman, Ethan Turok, Hantao Yu, Hengzhi Zhang

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Author Details

Josh Alman
  • Columbia University, New York, NY, USA
Ethan Turok
  • Columbia University, New York, NY, USA
Hantao Yu
  • Columbia University, New York, NY, USA
Hengzhi Zhang
  • Columbia University, New York, NY, USA

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Josh Alman, Ethan Turok, Hantao Yu, and Hengzhi Zhang. Tensor Ranks and the Fine-Grained Complexity of Dynamic Programming. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Generalizing work of Künnemann, Paturi, and Schneider [ICALP 2017], we study a wide class of high-dimensional dynamic programming (DP) problems in which one must find the shortest path between two points in a high-dimensional grid given a tensor of transition costs between nodes in the grid. This captures many classical problems which are solved using DP such as the knapsack problem, the airplane refueling problem, and the minimal-weight polygon triangulation problem. We observe that for many of these problems, the tensor naturally has low tensor rank or low slice rank. We then give new algorithms and a web of fine-grained reductions to tightly determine the complexity of these problems. For instance, we show that a polynomial speedup over the DP algorithm is possible when the tensor rank is a constant or the slice rank is 1, but that such a speedup is impossible if the tensor rank is slightly super-constant (assuming SETH) or the slice rank is at least 3 (assuming the APSP conjecture). We find that this characterizes the known complexities for many of these problems, and in some cases leads to new faster algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic programming
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Algebraic complexity theory
  • Fine-grained complexity
  • Dynamic programming
  • Least-weight subsequence


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  1. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for lcs and other sequence similarity measures. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 59-78. IEEE, 2015. Google Scholar
  2. Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean minimum spanning trees and bichromatic closest pairs. In Proceedings of the Sixth Annual Symposium on Computational Geometry, SCG '90, pages 203-210, 1990. Google Scholar
  3. Josh Alman. Limits on the universal method for matrix multiplication. Theory Of Computing, 17(1):1-30, 2021. Google Scholar
  4. Josh Alman, Ethan Turok, Hantao Yu, and Hengzhi Zhang. Tensor ranks and the fine-grained complexity of dynamic programming, 2023. URL: https://arxiv.org/abs/2309.04683.
  5. Josh Alman and Virginia Vassilevska Williams. Limits on all known (and some unknown) approaches to matrix multiplication. SIAM Journal on Computing, 0(0):FOCS18-285, 2021. Google Scholar
  6. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless seth is false). In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 51-58, 2015. Google Scholar
  7. Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 457-466. IEEE, 2016. Google Scholar
  8. Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. In Computing in Euclidean geometry, pages 47-123. World Scientific, 1995. Google Scholar
  9. Markus Bläser, Christian Ikenmeyer, Vladimir Lysikov, Anurag Pandey, and Frank-Olaf Schreyer. On the orbit closure containment problem and slice rank of tensors. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2565-2584. SIAM, 2021. Google Scholar
  10. Markus Bläser and Vladimir Lysikov. Slice rank of block tensors and irreversibility of structure tensors of algebras. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  11. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A Grochow, Eric Naslund, William F Sawin, and Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete Analysis, 3, 2017. Google Scholar
  12. Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless seth fails. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 661-670. IEEE, 2014. Google Scholar
  13. Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 79-97, 2015. Google Scholar
  14. Lijie Chen. On the hardness of approximate and exact (bichromatic) maximum inner product. Theory Of Computing, 16(4):1-50, 2020. Google Scholar
  15. Lijie Chen and Ryan Williams. An equivalence class for orthogonal vectors. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 21-40. SIAM, 2019. Google Scholar
  16. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. MIT Press, Cambridge, MA, USA, 2nd edition, 2001. Google Scholar
  17. Ernie Croot, Vsevolod F Lev, and Péter Pál Pach. Progression-free sets in are exponentially small. Annals of Mathematics, pages 331-337, 2017. Google Scholar
  18. Jordan S Ellenberg and Dion Gijswijt. On large subsets of with no three-term arithmetic progression. Annals of Mathematics, pages 339-343, 2017. Google Scholar
  19. Michael L Fredman. On computing the length of longest increasing subsequences. Discrete Mathematics, 11(1):29-35, 1975. Google Scholar
  20. Zvi Galil and Raffaele Giancarlo. Speeding up dynamic programming with applications to molecular biology. Theoretical Computer Science, 64(1):107-118, 1989. Google Scholar
  21. Edgar N Gilbert and Edward F Moore. Variable-length binary encodings. Bell System Technical Journal, 38(4):933-967, 1959. Google Scholar
  22. Sadashiva S Godbole. On efficient computation of matrix chain products. IEEE Transactions on Computers, 100(9):864-866, 1973. Google Scholar
  23. John Hershberger. An optimal visibility graph algorithm for triangulated simple polygons. Algorithmica, 4(1-4):141-155, 1989. Google Scholar
  24. D. S. Hirschberg and L. L. Larmore. The least weight subsequence problem. SIAM Journal on Computing, 16(4):628-638, 1987. URL: https://doi.org/10.1137/0216043.
  25. T.C. Hu and M. Shing. Combinatorial Algorithms: T.C. Hu and M.T. Shing. Dover Books on Computer Science Series. Dover Publications, 2002. Google Scholar
  26. TC Hu and MT Shing. Computation of matrix chain products. part i. SIAM Journal on Computing, 11(2):362-373, 1982. Google Scholar
  27. TC Hu and MT Shing. Computation of matrix chain products. part ii. SIAM Journal on Computing, 13(2):228-251, 1984. Google Scholar
  28. Russell Impagliazzo, Shachar Lovett, Ramamohan Paturi, and Stefan Schneider. 0-1 integer linear programming with a linear number of constraints. CoRR, abs/1401.5512, 2014. URL: https://arxiv.org/abs/1401.5512.
  29. Maria M. Klawe and Daniel J. Kleitman. An almost linear time algorithm for generalized matrix searching. SIAM Journal on Discrete Mathematics, 3(1):81-97, 1990. URL: https://doi.org/10.1137/0403009.
  30. DE Knuth. Optimum binary search trees. Acta Informatica, 1(1):14-25, 1971. Google Scholar
  31. Donald E Knuth and Michael F Plass. Breaking paragraphs into lines. Software: Practice and Experience, 11(11):1119-1184, 1981. Google Scholar
  32. Swastik Kopparty, Guy Moshkovitz, and Jeroen Zuiddam. Geometric rank of tensors and subrank of matrix multiplication. In 35th Computational Complexity Conference (CCC 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  33. Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the fine-grained complexity of one-dimensional dynamic programming. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  34. Thong Le and Dan Gusfield. Matrix chain multiplication and polygon triangulation revisited and generalized. arXiv preprint, 2021. URL: https://arxiv.org/abs/2104.01777.
  35. László Miklós Lovász and Lisa Sauermann. A lower bound for the k-multicolored sum-free problem in z mn. Proceedings of the London Mathematical Society, 119(1):55-103, 2019. Google Scholar
  36. Jiří Matoušek. Efficient partition trees. In Proceedings of the Seventh Annual Symposium on Computational Geometry, pages 1-9, New York, NY, USA, 1991. Association for Computing Machinery. Google Scholar
  37. Webb Miller and Eugene W. Myers. Sequence comparison with concave weighting functions. Bulletin of Mathematical Biology, 50(2):97-120, 1988. Google Scholar
  38. Atul Narkhede and Dinesh Manocha. Fast polygon triangulation based on seidel’s algorithm. In Graphics Gems V, pages 394-397. Elsevier, 1995. Google Scholar
  39. Eric Naslund and Will Sawin. Upper bounds for sunflower-free sets. In Forum of Mathematics, Sigma, volume 5, page e15. Cambridge University Press, 2017. Google Scholar
  40. Giorgio Ottaviani and Philipp Reichenbach. Tensor rank and complexity. arXiv preprint, 2020. URL: https://arxiv.org/abs/2004.01492.
  41. Muzafer Saracevic, Predrag Stanimirovic, Sead Mašovic, and Enver Biševac. Implementation of the convex polygon triangulation algorithm. Facta Universitatis, series: Mathematics and Informatics, 27(2):213-228, 2012. Google Scholar
  42. Will Sawin. Bounds for matchings in nonabelian groups. The Electronic Journal of Combinatorics, 25(4):4-23, 2018. Google Scholar
  43. Richard P Stanley. Catalan numbers. Cambridge University Press, 2015. Google Scholar
  44. Terence Tao. A symmetric formulation of the crootlev-pach-ellenberg-gijswijt capset bound. Tao’s blog post, 2016. Google Scholar
  45. Terence Tao and Will Sawin. Notes on the “slice rank” of tensors. Tao’s blog post, 2016. Google Scholar
  46. Robert Wilber. The concave least-weight subsequence problem revisited. Journal of Algorithms, 9(3):418-425, 1988. Google Scholar
  47. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2):357-365, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALP-A 2004). Google Scholar
  48. Ryan Williams. On the difference between closest, furthest, and orthogonal pairs: Nearly-linear vs barely-subquadratic complexity. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1207-1215. SIAM, 2018. Google Scholar
  49. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5), August 2018. Google Scholar
  50. Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput., 11(4):721-736, November 1982. Google Scholar
  51. F. Frances Yao. Efficient dynamic programming using quadrangle inequalities. In Symposium on the Theory of Computing, 1980. URL: https://api.semanticscholar.org/CorpusID:3154717.
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