Document

**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits.
Debordering is at the heart of understanding the difference between Valiant’s determinant vs permanent conjecture, and Mulmuley and Sohoni’s variation which uses border determinantal complexity. The debordering of matrix multiplication tensors by Bini played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Currently, very few debordering results are known.
In this work, we study the question of debordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures in the context of invariant theory, algebraic geometry, and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only linear in the degree. All previous known results were exponential in the degree. For polynomials with constant border Waring rank, our results imply an upper bound on the Waring rank linear in degree, which previously was only known for polynomials with border Waring rank at most 5.

Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Fixed-Parameter Debordering of Waring Rank. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.STACS.2024.30, author = {Dutta, Pranjal and Gesmundo, Fulvio and Ikenmeyer, Christian and Jindal, Gorav and Lysikov, Vladimir}, title = {{Fixed-Parameter Debordering of Waring Rank}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.30}, URN = {urn:nbn:de:0030-drops-197403}, doi = {10.4230/LIPIcs.STACS.2024.30}, annote = {Keywords: border complexity, Waring rank, debordering, apolarity} }

Document

**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

The celebrated result by Ben-Or and Cleve [SICOMP92] showed that algebraic formulas are polynomially equivalent to width-3 algebraic branching programs (ABP) for computing polynomials. i.e., VF = VBP₃. Further, there are simple polynomials, such as ∑_{i = 1}⁸ x_i y_i, that cannot be computed by width-2 ABPs [Allender and Wang, CC16]. Bringmann, Ikenmeyer and Zuiddam, [JACM18], on the other hand, studied these questions in the setting of approximate (i.e., border complexity) computation, and showed the universality of border width-2 ABPs, over fields of characteristic ≠ 2. In particular, they showed that polynomials that can be approximated by formulas can also be approximated (with only a polynomial blowup in size) by width-2 ABPs, i.e., VF ̅ = VBP₂ ̅. The power of border width-2 algebraic branching programs when the characteristic of the field is 2 was left open.
In this paper, we show that width-2 ABPs can approximate every polynomial irrespective of the field characteristic. We show that any polynomial f with 𝓁 monomials and with at most t odd-power indeterminates per monomial can be approximated by 𝒪(𝓁⋅ (deg(f)+2^t))-size width-2 ABPs. Since 𝓁 and t are finite, this proves universality of border width-2 ABPs. For univariate polynomials, we improve this upper-bound from O(deg(f)²) to O(deg(f)).
Moreover, we show that, if a polynomial f can be approximated by small formulas, then the polynomial f^d, for some small power d, can be approximated by small width-2 ABPs. Therefore, even over fields of characteristic two, border width-2 ABPs are a reasonably powerful computational model. Our construction works over any field.

Pranjal Dutta, Christian Ikenmeyer, Balagopal Komarath, Harshil Mittal, Saraswati Girish Nanoti, and Dhara Thakkar. On the Power of Border Width-2 ABPs over Fields of Characteristic 2. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.STACS.2024.31, author = {Dutta, Pranjal and Ikenmeyer, Christian and Komarath, Balagopal and Mittal, Harshil and Nanoti, Saraswati Girish and Thakkar, Dhara}, title = {{On the Power of Border Width-2 ABPs over Fields of Characteristic 2}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.31}, URN = {urn:nbn:de:0030-drops-197419}, doi = {10.4230/LIPIcs.STACS.2024.31}, annote = {Keywords: Algebraic branching programs, border complexity, characteristic 2} }

Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study algebraic complexity classes and their complete polynomials under homogeneous linear projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the first complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms.
Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Homogeneous Algebraic Complexity Theory and Algebraic Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.ITCS.2024.43, author = {Dutta, Pranjal and Gesmundo, Fulvio and Ikenmeyer, Christian and Jindal, Gorav and Lysikov, Vladimir}, title = {{Homogeneous Algebraic Complexity Theory and Algebraic Formulas}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {43:1--43:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.43}, URN = {urn:nbn:de:0030-drops-195713}, doi = {10.4230/LIPIcs.ITCS.2024.43}, annote = {Keywords: Homogeneous polynomials, Waring rank, Arithmetic formulas, Border complexity, Geometric Complexity theory, Symmetric polynomials} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ^[k] Π Σ ∧) and sum-product-of-constant-degree-polynomials (Σ^[k] Π Σ Π^[δ]), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha,Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work.
We give the first polynomial-time PIT for Σ^[k] Π Σ ∧. Further, we give the first quasipolynomial time blackbox PIT for both Σ^[k] Π Σ ∧ and Σ^[k] Π Σ Π^[δ]. No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.

Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 11:1-11:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.CCC.2021.11, author = {Dutta, Pranjal and Dwivedi, Prateek and Saxena, Nitin}, title = {{Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {11:1--11:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.11}, URN = {urn:nbn:de:0030-drops-142857}, doi = {10.4230/LIPIcs.CCC.2021.11}, annote = {Keywords: Polynomial identity testing, hitting set, depth-4 circuits} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Given polynomials f,g,h ∈ 𝔽[x₁,…,x_n] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s,deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen’s classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)).
Further, we show that any univariate polynomial family (f_d)_d, defined by the initial segment of the power series expansion of rational function g_d(x)/h_d(x) up to degree d (i.e. f_d = g_d/h_d od x^{d+1}), where circuit size of g is s_d and degree of g_d is at most d, can be computed by a circuit of size poly(s_d,deg(h_d),log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω (d)), assuming hardness of the integer factorization problem.
Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.

Pranjal Dutta, Gorav Jindal, Anurag Pandey, and Amit Sinhababu. Arithmetic Circuit Complexity of Division and Truncation. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 25:1-25:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.CCC.2021.25, author = {Dutta, Pranjal and Jindal, Gorav and Pandey, Anurag and Sinhababu, Amit}, title = {{Arithmetic Circuit Complexity of Division and Truncation}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {25:1--25:36}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.25}, URN = {urn:nbn:de:0030-drops-142990}, doi = {10.4230/LIPIcs.CCC.2021.25}, annote = {Keywords: Arithmetic Circuits, Division, Truncation, Division elimination, Rational function, Algebraic power series, Transcendental power series, Integer factorization} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

For a polynomial f, we study the sum of squares representation (SOS), i.e. f = ∑_{i ∈ [s]} c_i f_i² , where c_i are field elements and the f_i’s are polynomials. The size of the representation is the number of monomials that appear across the f_i’s. Its minimum is the support-sum S(f) of f.
For simplicity of exposition, we consider univariate f. A trivial lower bound for the support-sum of, a full-support univariate polynomial, f of degree d is S(f) ≥ d^{0.5}. We show that the existence of an explicit polynomial f with support-sum just slightly larger than the trivial bound, that is, S(f) ≥ d^{0.5+ε(d)}, for a sub-constant function ε(d) > ω(√{log log d/log d}), implies that VP ≠ VNP. The latter is a major open problem in algebraic complexity. A further consequence is that blackbox-PIT is in SUBEXP. Note that a random polynomial fulfills the condition, as there we have S(f) = Θ(d).
We also consider the sum-of-cubes representation (SOC) of polynomials. In a similar way, we show that here, an explicit hard polynomial even implies that blackbox-PIT is in P.

Pranjal Dutta, Nitin Saxena, and Thomas Thierauf. A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{dutta_et_al:LIPIcs.ITCS.2021.23, author = {Dutta, Pranjal and Saxena, Nitin and Thierauf, Thomas}, title = {{A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {23:1--23:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.23}, URN = {urn:nbn:de:0030-drops-135629}, doi = {10.4230/LIPIcs.ITCS.2021.23}, annote = {Keywords: VP, VNP, hitting set, circuit, polynomial, sparsity, SOS, SOC, PIT, lower bound} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail