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Border Complexity of Symbolic Determinant Under Rank One Restriction

Authors Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj

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Abhranil Chatterjee
  • ACM Unit, Indian Statistical Institute, Kolkata, India
Sumanta Ghosh
  • Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
Rohit Gurjar
  • Department of Computer Science and Engineering, IIT Bombay, India
Roshan Raj
  • Department of Computer Science and Engineering, IIT Bombay, India

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Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj. Border Complexity of Symbolic Determinant Under Rank One Restriction. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 2:1-2:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where the size of each A_i is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation i.e. whether VBP = VBP^ ̅. The power of approximation is well understood for some restricted models of computation, e.g. the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Markus Bläser et al., 2020], whereas the approximative closure of the last one captures the entire class of polynomial families computable by polynomial-sized formulas [Bringmann et al., 2017]. In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where for each 1 ≤ i ≤ n, A_i is of rank one. This class has been studied extensively [Edmonds, 1968; Jack Edmonds, 1979; Murota, 1993] and efficient identity testing algorithms are known for it [Lovász, 1989; Rohit Gurjar and Thomas Thierauf, 2020]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Plücker embedding of) a Grassmannian variety is closed.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Border Complexity
  • Symbolic Determinant
  • Valuated Matroid


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  1. Eric Allender and Fengming Wang. On the power of algebraic branching programs of width two. computational complexity, 25:217-253, March 2016. URL:
  2. Matthew Anderson, Amir Shpilka, and Ben Lee Volk. Personal communication, 2016. Google Scholar
  3. N. R. Aravind and Pushkar S. Joglekar. On the expressive power of read-once determinants. In Adrian Kosowski and Igor Walukiewicz, editors, Fundamentals of Computation Theory - 20th International Symposium, FCT 2015, Gdańsk, Poland, August 17-19, 2015, Proceedings, volume 9210 of Lecture Notes in Computer Science, pages 95-105. Springer, 2015. URL:
  4. Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. Algebraic branching programs, border complexity, and tangent spaces. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 21:1-21:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  5. Cristiano Bocci and Enrico Carlini. Hadamard products of hypersurfaces. Journal of Pure and Applied Algebra, 226(11):107078, 2022. URL:
  6. Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. On algebraic branching programs of small width. Journal of the ACM, 65, February 2017. URL:
  7. Peter Bürgisser. The complexity of factors of multivariate polynomials. Found. Comput. Math., 4(4):369-396, November 2004. Google Scholar
  8. Andreas W.M. Dress and Walter Wenzel. Valuated matroids: a new look at the greedy algorithm. Applied Mathematics Letters, 3(2):33-35, 1990. URL:
  9. Andreas W.M Dress and Walter Wenzel. Valuated matroids. Advances in Mathematics, 93(2):214-250, 1992. URL:
  10. Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Demystifying the border of depth-3 algebraic circuits. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 92-103, 2022. URL:
  11. Jack Edmonds. Systems of distinct representatives and linear algebra. Journal of research of the National Bureau of Standards, 71:241-245, 1967. Google Scholar
  12. Jack Edmonds. Matroid partition. Mathematics of the Decision Sciences, 11:335-345, 1968. Google Scholar
  13. Jack Edmonds. Matroid intersection. In P.L. Hammer, E.L. Johnson, and B.H. Korte, editors, Discrete Optimization I, volume 4 of Annals of Discrete Mathematics, pages 39-49. Elsevier, 1979. URL:
  14. András Frank. A weighted matroid intersection algorithm. Journal of Algorithms, 2(4):328-336, 1981. URL:
  15. Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in quasi-nc. Comput. Complex., 29(2):9, 2020. URL:
  16. Shaowei Lin and Bernd Sturmfels. Polynomial relations among principal minors of a 4×4-matrix. Journal of Algebra, 322(11):4121-4131, 2009. Computational Algebra. URL:
  17. László Lovász. Singular spaces of matrices and their application in combinatorics. Boletim da Sociedade Brasileira de Matemática, 20:87-99, October 1989. URL:
  18. 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:
  19. Ketan D. Mulmuley and Milind Sohoni. Geometric complexity theory i: An approach to the p vs. np and related problems. SIAM Journal on Computing, 31(2):496-526, 2001. URL:
  20. David Mumford. Algebraic Geometry I. Springer Berlin, Heidelberg, 1995. Google Scholar
  21. Kazuo Murota. Mixed matrices: Irreducibility and decomposition. In Richard A. Brualdi, Shmuel Friedland, and Victor Klee, editors, Combinatorial and Graph-Theoretical Problems in Linear Algebra, pages 39-71, New York, NY, 1993. Springer New York. Google Scholar
  22. Kazuo Murota. Valuated matroid intersection i: Optimality criteria. SIAM Journal on Discrete Mathematics, 9(4):545-561, 1996. URL:
  23. James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press, Inc., New York, NY, USA, 2006. Google Scholar
  24. 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:
  25. Alexander Schrijver. Combinatorial optimization : polyhedra and efficiency. Vol. B. , Matroids, trees, stable sets. chapters 39-69. Algorithms and combinatorics. Springer-Verlag, Berlin, Heidelberg, New York, N.Y., et al., 2003. URL:
  26. Jiang Zeng. A bijective proof of Muir’s identity and the Cauchy-Binet formula. Linear Algebra and its Applications, 184:79-82, 1993. Google Scholar
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