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Complete Volume

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

LIPIcs, Volume 284, FSTTCS 2023, Complete Volume

43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 1-784, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{bouyer_et_al:LIPIcs.FSTTCS.2023, title = {{LIPIcs, Volume 284, FSTTCS 2023, Complete Volume}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {1--784}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023}, URN = {urn:nbn:de:0030-drops-193724}, doi = {10.4230/LIPIcs.FSTTCS.2023}, annote = {Keywords: LIPIcs, Volume 284, FSTTCS 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Front Matter, Table of Contents, Preface, Conference Organization

43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2023.0, author = {Bouyer, Patricia and Srinivasan, Srikanth}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.0}, URN = {urn:nbn:de:0030-drops-193737}, doi = {10.4230/LIPIcs.FSTTCS.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We study the question of local testability of low (constant) degree functions from a product domain 𝒮_1 × … × 𝒮_n to a field 𝔽, where 𝒮_i ⊆ 𝔽 can be arbitrary constant sized sets. We show that this family is locally testable when the grid is "symmetric". That is, if 𝒮_i = 𝒮 for all i, there is a probabilistic algorithm using constantly many queries that distinguishes whether f has a polynomial representation of degree at most d or is Ω(1)-far from having this property. In contrast, we show that there exist asymmetric grids with |𝒮_1| = ⋯ = |𝒮_n| = 3 for which testing requires ω_n(1) queries, thereby establishing that even in the context of polynomials, local testing depends on the structure of the domain and not just the distance of the underlying code.
The low-degree testing problem has been studied extensively over the years and a wide variety of tools have been applied to propose and analyze tests. Our work introduces yet another new connection in this rich field, by building low-degree tests out of tests for "junta-degrees". A function f:𝒮_1 × ⋯ × 𝒮_n → 𝒢, for an abelian group 𝒢 is said to be a junta-degree-d function if it is a sum of d-juntas. We derive our low-degree test by giving a new local test for junta-degree-d functions. For the analysis of our tests, we deduce a small-set expansion theorem for spherical/hamming noise over large grids, which may be of independent interest.

Prashanth Amireddy, Srikanth Srinivasan, and Madhu Sudan. Low-Degree Testing over Grids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{amireddy_et_al:LIPIcs.APPROX/RANDOM.2023.41, author = {Amireddy, Prashanth and Srinivasan, Srikanth and Sudan, Madhu}, title = {{Low-Degree Testing over Grids}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {41:1--41:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.41}, URN = {urn:nbn:de:0030-drops-188665}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.41}, annote = {Keywords: Property testing, Low-degree testing, Small-set expansion, Local testing} }

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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula.
Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas.
This improves on previous results in the regime when d is small (i.e., d = s^o(1)). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting.
We also show that these results cannot be improved in the monotone setting, even for commutative formulas.

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, and Sébastien Tavenas. Towards Optimal Depth-Reductions for Algebraic Formulas. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fournier_et_al:LIPIcs.CCC.2023.28, author = {Fournier, Herv\'{e} and Limaye, Nutan and Malod, Guillaume and Srinivasan, Srikanth and Tavenas, S\'{e}bastien}, title = {{Towards Optimal Depth-Reductions for Algebraic Formulas}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {28:1--28:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.28}, URN = {urn:nbn:de:0030-drops-182986}, doi = {10.4230/LIPIcs.CCC.2023.28}, annote = {Keywords: Algebraic formulas, depth-reduction} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory.
In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials.
We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size.

Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan. On the VNP-Hardness of Some Monomial Symmetric Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{curticapean_et_al:LIPIcs.FSTTCS.2022.16, author = {Curticapean, Radu and Limaye, Nutan and Srinivasan, Srikanth}, title = {{On the VNP-Hardness of Some Monomial Symmetric Polynomials}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.16}, URN = {urn:nbn:de:0030-drops-174081}, doi = {10.4230/LIPIcs.FSTTCS.2022.16}, annote = {Keywords: algebraic complexity, symmetric polynomial, permanent, Sidon set} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

We study the following natural question on random sets of points in 𝔽₂^m:
Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ 𝔽₂^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z?
We show that, for r ≤ γ m (where γ > 0 is a small, absolute constant) and k = (1-ε)⋅binom(m, ≤ r) for any constant ε > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly binom(m, ≤ r) - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of binom(m, ≤ r) - binom(log₂ k, ≤ r) ≫ binom(m, ≤ r) - k.
Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ε > 0, we can recover from (1-ε)⋅binom(m, ≤ m-d-1) random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈ binom(m, ≤ m-(d/2)) random errors for the same range of parameters.

Siddharth Bhandari, Prahladh Harsha, Ramprasad Saptharishi, and Srikanth Srinivasan. Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhandari_et_al:LIPIcs.CCC.2022.31, author = {Bhandari, Siddharth and Harsha, Prahladh and Saptharishi, Ramprasad and Srinivasan, Srikanth}, title = {{Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.31}, URN = {urn:nbn:de:0030-drops-165934}, doi = {10.4230/LIPIcs.CCC.2022.31}, annote = {Keywords: Reed-Muller codes, polynomials, weight-distribution, vanishing ideals, erasures, capacity} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

We make progress on understanding a lower bound technique that was recently used by the authors to prove the first superpolynomial constant-depth circuit lower bounds against algebraic circuits.
More specifically, our previous work applied the well-known partial derivative method in a new setting, that of lopsided set-multilinear polynomials. A set-multilinear polynomial P ∈ 𝔽[X_1,…,X_d] (for disjoint sets of variables X_1,…,X_d) is a linear combination of monomials, each of which contains one variable from X_1,…,X_d. A lopsided space of set-multilinear polynomials is one where the sets X_1,…,X_d are allowed to have different sizes (we use the adjective "lopsided" to stress this feature). By choosing a suitable lopsided space of polynomials, and using a suitable version of the partial-derivative method for proving lower bounds, we were able to prove constant-depth superpolynomial set-multilinear formula lower bounds even for very low-degree polynomials (as long as d is a growing function of the number of variables N). This in turn implied lower bounds against general formulas of constant-depth.
A priori, there is nothing stopping these techniques from giving us lower bounds against algebraic formulas of any depth. We investigate the extent to which this lower bound can extend to greater depths. We prove the following results.
1) We observe that our choice of the lopsided space and the kind of partial-derivative method used can be modeled as the choice of a multiset W ⊆ [-1,1] of size d. Our first result completely characterizes, for any product-depth Δ, the best lower bound we can prove for set-multilinear formulas of product-depth Δ in terms of some combinatorial properties of W, that we call the depth-Δ tree bias of W.
2) We show that the maximum depth-3 tree bias, over multisets W of size d, is Θ(d^{1/4}). This shows a stronger formula lower bound of N^{Ω(d^{1/4})} for set-multilinear formulas of product-depth 3, and also puts a non-trivial constraint on the best lower bounds we can hope to prove at this depth in this framework (a priori, we could have hoped to prove a lower bound of N^{Ω(Δ d^{1/Δ})} at product-depth Δ).
3) Finally, we show that for small Δ, our proof technique cannot hope to prove lower bounds of the form N^{Ω(d^{1/poly(Δ)})}. This seems to strongly hint that new ideas will be required to prove lower bounds for formulas of unbounded depth.

Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 32:1-32:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{limaye_et_al:LIPIcs.CCC.2022.32, author = {Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien}, title = {{On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {32:1--32:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.32}, URN = {urn:nbn:de:0030-drops-165942}, doi = {10.4230/LIPIcs.CCC.2022.32}, annote = {Keywords: Partial Derivative Method, Barriers to Lower Bounds} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}ⁿ → {0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x₁,…,x_n), has degree at least log n - O(log log n). This was improved to a tight (log n - O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)).
In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)^{1/2-o(1)} and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)).
Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)^c, then the minimum possible probabilistic degree of such an f is at least (log n)^{c/(c+1)-o(1)}, and we show this is tight up to (log n)^{o(1)} factors.

Srikanth Srinivasan and S. Venkitesh. On the Probabilistic Degree of an n-Variate Boolean Function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{srinivasan_et_al:LIPIcs.APPROX/RANDOM.2021.42, author = {Srinivasan, Srikanth and Venkitesh, S.}, title = {{On the Probabilistic Degree of an n-Variate Boolean Function}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {42:1--42:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.42}, URN = {urn:nbn:de:0030-drops-147356}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.42}, annote = {Keywords: truly n-variate, Boolean function, probabilistic polynomial, probabilistic degree} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Stanley, 1999], in Schubert calculus [Ledoux and Malham, 2010] as well as in Enumerative combinatorics [Gasharov, 1996; Stanley, 1984; Stanley, 1999]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [Hallgren et al., 2000; Ryan O'Donnell and John Wright, 2015] and Geometric complexity theory [Ikenmeyer and Panova, 2017].
However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas.
As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.

Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, and Srikanth Srinivasan. Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 14:1-14:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chaugule_et_al:LIPIcs.CCC.2020.14, author = {Chaugule, Prasad and Kumar, Mrinal and Limaye, Nutan and Mohapatra, Chandra Kanta and She, Adrian and Srinivasan, Srikanth}, title = {{Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {14:1--14:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.14}, URN = {urn:nbn:de:0030-drops-125660}, doi = {10.4230/LIPIcs.CCC.2020.14}, annote = {Keywords: Schur polynomial, Jacobian, Algebraic independence, Generalized Vandermonde determinant, Taylor expansion, Formula complexity, Lower bound} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

In this paper we prove two results about AC^0[oplus] circuits.
(1) We show that for d(N) = o(sqrt(log N/log log N)) and N <= s(N) <= 2^(dN^(1/4d^2)) there is an explicit family of functions {f_N:{0,1}^N - > {0,1}} such that
- f_N has uniform AC^0 formulas of depth d and size at most s;
- f_N does not have AC^0[oplus] formulas of depth d and size s^epsilon, where epsilon is a fixed absolute constant.
This gives a quantitative improvement on the recent result of Limaye, Srinivasan, Sreenivasaiah, Tripathi, and Venkitesh, (STOC, 2019), which proved a similar Fixed-Depth Size-Hierarchy theorem but for d << log log N and s << exp(N^(1/2^Omega(d))).
As in the previous result, we use the Coin Problem to prove our hierarchy theorem. Our main technical result is the construction of uniform size-optimal formulas for solving the coin problem with improved sample complexity (1/delta)^O(d) (down from (1/delta)^(2^O(d)) in the previous result).
(2) In our second result, we show that randomness buys depth in the AC^0[oplus] setting. Formally, we show that for any fixed constant d >= 2, there is a family of Boolean functions that has polynomial-sized randomized uniform AC^0 circuits of depth d but no polynomial-sized (deterministic) AC^0[oplus] circuits of depth d.
Previously Viola (Computational Complexity, 2014) showed that an increase in depth (by at least 2) is essential to avoid superpolynomial blow-up while derandomizing randomized AC^0 circuits. We show that an increase in depth (by at least 1) is essential even for AC^0[oplus].
As in Viola’s result, the separating examples are promise variants of the Majority function on N inputs that accept inputs of weight at least N/2 + N/(log N)^(d-1) and reject inputs of weight at most N/2 - N/(log N)^(d-1).

Nutan Limaye, Srikanth Srinivasan, and Utkarsh Tripathi. More on AC^0[oplus] and Variants of the Majority Function. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{limaye_et_al:LIPIcs.FSTTCS.2019.22, author = {Limaye, Nutan and Srinivasan, Srikanth and Tripathi, Utkarsh}, title = {{More on AC^0\lbrackoplus\rbrack and Variants of the Majority Function}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.22}, URN = {urn:nbn:de:0030-drops-115844}, doi = {10.4230/LIPIcs.FSTTCS.2019.22}, annote = {Keywords: AC^0\lbrackoplus\rbrack, Coin Problem, Promise Majority} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The probabilistic degree of a Boolean function f:{0,1}^n -> {0,1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions - specifically symmetric Boolean functions - have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems.
In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. On the Probabilistic Degrees of Symmetric Boolean Functions. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{srinivasan_et_al:LIPIcs.FSTTCS.2019.28, author = {Srinivasan, Srikanth and Tripathi, Utkarsh and Venkitesh, S.}, title = {{On the Probabilistic Degrees of Symmetric Boolean Functions}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {28:1--28:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.28}, URN = {urn:nbn:de:0030-drops-115908}, doi = {10.4230/LIPIcs.FSTTCS.2019.28}, annote = {Keywords: Symmetric Boolean function, probabilistic degree} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades.
Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.

Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity Helps to Compute Majority. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.23, author = {Oliveira, Igor Carboni and Santhanam, Rahul and Srinivasan, Srikanth}, title = {{Parity Helps to Compute Majority}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.23}, URN = {urn:nbn:de:0030-drops-108453}, doi = {10.4230/LIPIcs.CCC.2019.23}, annote = {Keywords: Computational Complexity, Boolean Circuits, Lower Bounds, Parity, Majority} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We show that there is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time.
Formally, for any constants d,k, there is an epsilon > 0 such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has size at most n^{1+epsilon}. The algorithm runs in time 2^{n-n^{Omega(epsilon)}}.
Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit of linear size).
The main new tool is the use of a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett, Moran and Zhang (FOCS 2017). We show that their ideas fit nicely into a memoization approach that yields the #SAT algorithms.

Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bajpai_et_al:LIPIcs.ITCS.2019.8, author = {Bajpai, Swapnam and Krishan, Vaibhav and Kush, Deepanshu and Limaye, Nutan and Srinivasan, Srikanth}, title = {{A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.8}, URN = {urn:nbn:de:0030-drops-101010}, doi = {10.4230/LIPIcs.ITCS.2019.8}, annote = {Keywords: SAT, Polynomial Threshold Functions, Constant-depth Boolean Circuits, Linear Decision Trees, Zero-error randomized algorithms} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We study the probabilistic degree over R of the OR function on n variables. For epsilon in (0,1/3), the epsilon-error probabilistic degree of any Boolean function f:{0,1}^n -> {0,1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials Pol in R[x_1,...,x_n] entirely supported on polynomials of degree at most d such that for all z in {0,1}^n, we have Pr_{P ~ Pol}[P(z) = f(z)] >= 1- epsilon. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the epsilon-error probabilistic degree of the OR function is at most O(log n * log(1/epsilon)). Our first observation is that this can be improved to O{log (n atop <= log(1/epsilon))}, which is better for small values of epsilon.
In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution Pol have the following special structure: P(x_1,...,x_n) = 1 - prod_{i in [t]} (1- L_i(x_1,...,x_n)), where each L_i(x_1,..., x_n) is a linear form in the variables x_1,...,x_n, i.e., the polynomial 1-P(bar{x}) is a product of affine forms. We show that the epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Omega(log (n over <= log(1/epsilon))/log^2 (log (n over <= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors).

Siddharth Bhandari, Prahladh Harsha, Tulasimohan Molli, and Srikanth Srinivasan. On the Probabilistic Degree of OR over the Reals. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bhandari_et_al:LIPIcs.FSTTCS.2018.5, author = {Bhandari, Siddharth and Harsha, Prahladh and Molli, Tulasimohan and Srinivasan, Srikanth}, title = {{On the Probabilistic Degree of OR over the Reals}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {5:1--5:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.5}, URN = {urn:nbn:de:0030-drops-99044}, doi = {10.4230/LIPIcs.FSTTCS.2018.5}, annote = {Keywords: Polynomials over reals, probabilistic polynomials, probabilistic degree, OR polynomial} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.

Ninad Rajgopal, Rahul Santhanam, and Srikanth Srinivasan. Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{rajgopal_et_al:LIPIcs.MFCS.2018.78, author = {Rajgopal, Ninad and Santhanam, Rahul and Srinivasan, Srikanth}, title = {{Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {78:1--78:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.78}, URN = {urn:nbn:de:0030-drops-96607}, doi = {10.4230/LIPIcs.MFCS.2018.78}, annote = {Keywords: circuit satisfiability, circuit lower bounds, polynomial method, derandomization} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show explicit separations between the expressive powers of multilinear formulas of small-depth and all polynomial sizes.
Formally, for any s = s(n) = n^{O(1)} and any delta>0, we construct explicit families of multilinear polynomials P_n in F[x_1,...,x_n] that have multilinear formulas of size s and depth three but no multilinear formulas of size s^{1/2-delta} and depth o(log n/log log n).
As far as we know, this is the first such result for an algebraic model of computation.
Our proof can be viewed as a derandomization of a lower bound technique of Raz (JACM 2009) using epsilon-biased spaces.

Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chillara_et_al:LIPIcs.ICALP.2018.36, author = {Chillara, Suryajith and Limaye, Nutan and Srinivasan, Srikanth}, title = {{A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.36}, URN = {urn:nbn:de:0030-drops-90401}, doi = {10.4230/LIPIcs.ICALP.2018.36}, annote = {Keywords: Algebraic circuit complexity, Multilinear formulas, Lower Bounds} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within P. In this paper, we study the algebraic formula complexity of multiplying d many 2x2 matrices, denoted IMM_d, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear.
Formally, for each depth Delta <= log d, we show that any product-depth Delta multilinear formula for IMM_d must have size exp(Omega(Delta d^{1/Delta})). It also follows from this that any multilinear circuit of product-depth Delta for the same polynomial of the above form must have a size of exp(Omega(d^{1/Delta})). In particular, any polynomial-sized multilinear formula for IMM_d must have depth Omega(log d), and any polynomial-sized multilinear circuit for IMM_d must have depth Omega(log d/log log d). Both these bounds are tight up to constant factors.
Our lower bound has the following consequences for multilinear formula complexity.
Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size s can be converted to one of size s^{O(1)} and depth O(log s); further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to o(log s) without a superpolynomial blow-up in size.
Separations from general formulas: Shpilka and Yehudayoff (FnTTCS 2010) asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a non-trivial upper bound for IMM_d implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size s and product-depth Delta = o(log s), general formulas of size s and product-depth Delta cannot be converted to multilinear formulas of size s^{O(1)} and product-depth Delta, when the underlying field has characteristic zero.

Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chillara_et_al:LIPIcs.STACS.2018.21, author = {Chillara, Suryajith and Limaye, Nutan and Srinivasan, Srikanth}, title = {{Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.21}, URN = {urn:nbn:de:0030-drops-85235}, doi = {10.4230/LIPIcs.STACS.2018.21}, annote = {Keywords: Algebraic circuit complexity, Multilinear formulas, Lower Bounds} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate polynomials of total degree at most d over grids, i.e. sets of the form A_1 \times A_2 \times \cdots \times A_n, form error-correcting codes (of distance at least 2^{-d} provided \min_i\{|A_i|\}\geq 2). In this work we explore their local decodability and local testability. While these aspects have been studied extensively when A_1 = \cdots = A_n = \F_q are the same finite field, the setting when A_i's are not the full field does not seem to have been explored before.
In this work we focus on the case A_i = {0,1} for every i. We show that for every field (finite or otherwise) there is a test whose query complexity depends only on the degree (and not on the number of variables). In contrast we show that decodability is possible over fields of positive characteristic (with query complexity growing with the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with $n$. As a consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable but not locally decodable.
Classical results on local decoding and testing of polynomials have relied on the 2-transitive symmetries of the space of low-degree polynomials (under affine transformations). Grids do not possess this symmetry: So we introduce some new techniques to overcome this handicap and in particular use the hypercontractivity of the (constant weight) noise operator on the Hamming cube.

Srikanth Srinivasan and Madhu Sudan. Local Decoding and Testing of Polynomials over Grids. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{srinivasan_et_al:LIPIcs.ITCS.2018.26, author = {Srinivasan, Srikanth and Sudan, Madhu}, title = {{Local Decoding and Testing of Polynomials over Grids}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {26:1--26:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.26}, URN = {urn:nbn:de:0030-drops-83294}, doi = {10.4230/LIPIcs.ITCS.2018.26}, annote = {Keywords: Property testing, Coding theory, Low-degree testing, Local decoding, Local testing} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring F<x_1,...,x_N>, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.
- We show explicit exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde, Malod, and Perifel (ECCC 2016), who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree.
- We show explicit exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye, Malod, and Srinivasan (Theory of Computing 2016) and the above lower bounds of Lagarde et al., which are known to be incomparable.
- We make progress on a question of Nisan (STOC 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of n^{Omega(log d)} for any UPT formula computing the product of d n*n matrices.
When d <= log n, we can also prove superpolynomial lower bounds for formulas with up to 2^{o(d)} many parse trees (for computing the same polynomial). Improving this bound to allow for 2^{O(d)} trees would yield an unconditional separation between ABPs and Formulas.
- We give deterministic white-box PIT algorithms for UPT circuits over any field (strengthening a result of Lagarde et al. (2016)) and also for sums of a constant number of UPT circuits with different parse trees.

Guillaume Lagarde, Nutan Limaye, and Srikanth Srinivasan. Lower Bounds and PIT for Non-Commutative Arithmetic Circuits with Restricted Parse Trees. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lagarde_et_al:LIPIcs.MFCS.2017.41, author = {Lagarde, Guillaume and Limaye, Nutan and Srinivasan, Srikanth}, title = {{Lower Bounds and PIT for Non-Commutative Arithmetic Circuits with Restricted Parse Trees}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.41}, URN = {urn:nbn:de:0030-drops-81094}, doi = {10.4230/LIPIcs.MFCS.2017.41}, annote = {Keywords: Non-commutative Arithemetic circuits, Partial derivatives, restrictions} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC^0[\oplus] basis (unbounded fan-in AND, OR, NOT and MOD_2 gates). We show, for all d(n) <= O(log n/log log n), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size n^{o(d)} (moreover, this is optimal in that n^{o(d)} cannot be improved to n^{O(d)}). This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions {0,1}^n to {0,1} that agree with the Majority function on 3/4 fraction of inputs.
AC^0[\oplus] formula lower bound.
We show that every depth-d AC^0[\oplus] formula of size s has a (1/8)-error polynomial approximation over F_2 of degree O((log s)/d)^{d-1}. This strengthens a classic $O(log s)^{d-1}$ degree approximation for circuits due to Razborov. Since the Majority function has approximate degree Theta(\sqrt n), this result implies an \exp(\Omega(dn^{1/2(d-1)})) lower bound on the depth-d AC^0[\oplus] formula size of all Approximate Majority functions for all d(n) <= O(log n).
Monotone AC^0 circuit upper bound.
For all d(n) <= O(log n/log log n), we give a randomized construction of depth-d monotone AC^0 circuits (without NOT or MOD_2 gates) of size \exp(O(n^{1/2(d-1)}))} that compute an Approximate Majority function. This strengthens a construction of formulas of size \exp(O(dn^{1/2(d-1)})) due to Amano.

Benjamin Rossman and Srikanth Srinivasan. Separation of AC^0[oplus] Formulas and Circuits. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rossman_et_al:LIPIcs.ICALP.2017.50, author = {Rossman, Benjamin and Srinivasan, Srikanth}, title = {{Separation of AC^0\lbrackoplus\rbrack Formulas and Circuits}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.50}, URN = {urn:nbn:de:0030-drops-73904}, doi = {10.4230/LIPIcs.ICALP.2017.50}, annote = {Keywords: circuit complexity, lower bounds, approximate majority, polynomial method} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -> {0,1}, define its k-lift to be F_k:{0,1}^n -> {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as \gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n].
Our results are the following:
* Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) >= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) <= 1/2+o(1) - as small as possible - but gamma_{d,3}(F) >= 1/2 + Omega(1).
* Increasing k doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) <= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z.
We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.

Abhishek Bhrushundi, Prahladh Harsha, and Srikanth Srinivasan. On Polynomial Approximations Over Z/2^kZ*. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bhrushundi_et_al:LIPIcs.STACS.2017.12, author = {Bhrushundi, Abhishek and Harsha, Prahladh and Srinivasan, Srikanth}, title = {{On Polynomial Approximations Over Z/2^kZ*}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {12:1--12:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.12}, URN = {urn:nbn:de:0030-drops-70212}, doi = {10.4230/LIPIcs.STACS.2017.12}, annote = {Keywords: Polynomials over rings, Approximation by polynomials, Boolean functions, Non-classical polynomials} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime.
Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial.
We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust.
We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,... , P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014).
One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest.

Prahladh Harsha and Srikanth Srinivasan. Robust Multiplication-Based Tests for Reed-Muller Codes. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.FSTTCS.2016.17, author = {Harsha, Prahladh and Srinivasan, Srikanth}, title = {{Robust Multiplication-Based Tests for Reed-Muller Codes}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {17:1--17:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.17}, URN = {urn:nbn:de:0030-drops-68524}, doi = {10.4230/LIPIcs.FSTTCS.2016.17}, annote = {Keywords: Polynomials over finite fields, Schwartz-Zippel lemma, Low degree testing, Low degree long code} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We make progress on some questions related to polynomial approximations of AC^0. It is known, from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC^0 circuit of size s and depth d has an epsilon-error probabilistic polynomial over the reals of degree (log (s/epsilon))^{O(d)}. We improve this upper bound to (log s)^{O(d)}* log(1/epsilon), which is much better for small values of epsilon.
We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)^{O(d)}* log(1/epsilon)-wise independence fools AC^0, improving on Tal's strengthening of Braverman's theorem (J. ACM 2010) that (log (s/epsilon))^{O(d)}-wise independence fools AC^0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC^0 that achieves optimal dependence on the error epsilon.
We also prove lower bounds on the best polynomial approximations to AC^0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least ~Omega(sqrt{log n}). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015).

Prahladh Harsha and Srikanth Srinivasan. On Polynomial Approximations to AC^0. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.APPROX-RANDOM.2016.32, author = {Harsha, Prahladh and Srinivasan, Srikanth}, title = {{On Polynomial Approximations to AC^0}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.32}, URN = {urn:nbn:de:0030-drops-66550}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.32}, annote = {Keywords: Constant-depth Boolean circuits, Polynomials over reals, pseudo-random generators, k-wise independence} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is epsilon_d > 0 such that Parity has correlation at most 1/n^{Omega(1)} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires, and the Generalized Andreev Function has correlation at most 1/2^{n^{Omega(1)}} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires. Previously, only worst-case lower bounds in this setting were known [Impagliazzo/Paturi/Saks, SIAM J. Comp., 1997].
We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth-$d$ threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC^0 circuits with n^{o(1)} general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC^0 with n^{o(1)} threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas ofany depth.
Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.

Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan. Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 1:1-1:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chen_et_al:LIPIcs.CCC.2016.1, author = {Chen, Ruiwen and Santhanam, Rahul and Srinivasan, Srikanth}, title = {{Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {1:1--1:35}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.1}, URN = {urn:nbn:de:0030-drops-58447}, doi = {10.4230/LIPIcs.CCC.2016.1}, annote = {Keywords: threshold circuit, satisfiability algorithm, circuit lower bound} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In this paper, we address the question of whether the recent derandomization results obtained by the use of the low-degree long code can be extended to other product settings. We consider two settings: (1) the graph product results of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of K_3^{\otimes R} which exhibits the following property
(shown for K_3^{\otimes R} by Alon et al.): independent sets close in
size to the maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur
and Shinkar shows that if there exist two sets of vertices A and B
in K_3^{\otimes R} with very few edges with one endpoint in A and
another in B, then it must be the case that the two sets A and B
share a single influential coordinate. In our second result, we show
that a similar "majority is stablest" statement holds good for a
considerably smaller subgraph of K_3^{\otimes R}. Furthermore using
this result, we give a more efficient reduction from Unique Games
to the graph coloring problem, leading to improved hardness of
approximation results for coloring.

Irit Dinur, Prahladh Harsha, Srikanth Srinivasan, and Girish Varma. Derandomized Graph Product Results Using the Low Degree Long Code. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 275-287, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dinur_et_al:LIPIcs.STACS.2015.275, author = {Dinur, Irit and Harsha, Prahladh and Srinivasan, Srikanth and Varma, Girish}, title = {{Derandomized Graph Product Results Using the Low Degree Long Code}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {275--287}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.275}, URN = {urn:nbn:de:0030-drops-49200}, doi = {10.4230/LIPIcs.STACS.2015.275}, annote = {Keywords: graph product, derandomization, low degree long code, graph coloring} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

A polynomial P in F[X_1,...,X_n] is said to epsilon-approximate a
boolean function F:{0,1}^n -> {0,1} under distribution D over {0,1}^n
if for a random x chosen according to distribution D, the probability
that P(x) is not equal to F(x) is at most epsilon. Smolensky (1987)
showed that for any constant distinct primes p and q, any polynomial P
in F_p[x_1,...,x_n] that (1/2q - Omega(1))-approximates the boolean
function MOD_q:{0,1}^n->{0,1} -- which accepts its input iff the
number of ones is non-zero modulo q -- under the uniform distribution
must have degree Omega(n^{1/2}).
We consider the problem of finding an explicit function
f:{0,1}^n->{0,1} that has no epsilon-approximating polynomial of
degree less than n^{1/2 + Omega(1)} under *some distribution*, for
some constant epsilon>0. We show a number of negative results in this
direction: specifically, we show that many interesting classes of
functions including symmetric functions and linear threshold functions
do have approximating polynomials of degree O(n^{1/2+o(1)}) under
every distribution. This demonstrates the power of this model of
computation.
The above results, in turn, provide further motivation for this lower
bound question. Using the upper bounds obtained above, we show that
finding such a function f would have applications to: lower bounds for
AC^0 o F where F is the class of symmetric and threshold gates;
stronger lower bounds for 1-round compression by ACC^0[p] circuits;
improved correlation lower bounds against low degree polynomials; and
(under further conditions) showing that the Inner Product (over F_2)
function does not have small AC^0 o MOD_2 circuits.

Srikanth Srinivasan. On Improved Degree Lower Bounds for Polynomial Approximation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 201-212, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{srinivasan:LIPIcs.FSTTCS.2013.201, author = {Srinivasan, Srikanth}, title = {{On Improved Degree Lower Bounds for Polynomial Approximation}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {201--212}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.201}, URN = {urn:nbn:de:0030-drops-43737}, doi = {10.4230/LIPIcs.FSTTCS.2013.201}, annote = {Keywords: Polynomials, Approximation, Compression, Circuit lower bounds} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

In this paper, we introduce and develop the method of certifying polynomials for proving AC^0 circuit lower bounds.
We use this method to show that Approximate Majority cannot be computed by AC^0(parity) circuits of size n^{1 + o(1)}. This implies a separation between the power of AC^0(parity) circuits of
near-linear size and uniform AC^0(parity) (and even AC^0) circuits of polynomial size.
This also implies a separation between randomized AC^0(parity) circuits of linear size and deterministic AC^0(parity) circuits of near-linear size.
Our proof using certifying polynomials extends the deterministic restrictions technique of Chaudhuri and Radhakrishnan, who showed that Approximate Majority cannot be computed by AC^0 circuits of size n^{1+o(1)}.
At the technical level, we show that for every ACP circuit C of near-linear size, there is a low degree variety V over F_2 such that the restriction of C to V is constant.
We also prove other results exploring various aspects of the power of certifying polynomials. In the process, we show an essentially optimal lower bound of Omega\left(\log^{\Theta(d)} s \cdot \log \frac{1}{\epsilon} \right) on the degree of \epsilon-approximating polynomials for AC^0(parity) circuits of size s.

Swastik Kopparty and Srikanth Srinivasan. Certifying polynomials for AC^0(parity) circuits, with applications. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 36-47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kopparty_et_al:LIPIcs.FSTTCS.2012.36, author = {Kopparty, Swastik and Srinivasan, Srikanth}, title = {{Certifying polynomials for AC^0(parity) circuits, with applications}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {36--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.36}, URN = {urn:nbn:de:0030-drops-38467}, doi = {10.4230/LIPIcs.FSTTCS.2012.36}, annote = {Keywords: Constant-depth Boolean circuits, Polynomials over finite fields, Size hierarchies} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Using $\varepsilon$-bias spaces over $\F_2$, we show that the Remote Point Problem (RPP), introduced by Alon et al \cite{APY09}, has an $\NC^2$ algorithm (achieving the same parameters as \cite{APY09}). We study a generalization of the Remote Point Problem to groups: we replace $\F_2^n$ by $\mcG^n$ for an arbitrary fixed group $\mcG$. When $\mcG$ is Abelian we give an $\NC^2$ algorithm for RPP, again using $\varepsilon$-bias spaces. For nonabelian $\mcG$, we give a deterministic polynomial-time algorithm for RPP. We also show the connection to construction of expanding generator sets for the group $\mcG^n$. All our algorithms for the RPP achieve essentially the same parameters as \cite{APY09}.

Vikraman Arvind and Srikanth Srinivasan. The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 59-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2010.2444, author = {Arvind, Vikraman and Srinivasan, Srikanth}, title = {{The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {59--70}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2444}, URN = {urn:nbn:de:0030-drops-24449}, doi = {10.4230/LIPIcs.STACS.2010.2444}, annote = {Keywords: Small Bias Spaces, Expander Graphs, Cayley Graphs, Remote Point Problem} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Motivated by the Hadamard product of matrices we define the Hadamard
product of multivariate polynomials and study its arithmetic circuit
and branching program complexity. We also give applications and
connections to polynomial identity testing. Our main results are
the following.
\begin{itemize}
\item[$\bullet$] We show that noncommutative polynomial identity testing for
algebraic branching programs over rationals is complete for
the logspace counting class $\ceql$, and over fields of characteristic
$p$ the problem is in $\ModpL/\Poly$.
\item[$\bullet$] We show an exponential lower bound for expressing the
Raz-Yehudayoff polynomial as the Hadamard product of two monotone
multilinear polynomials. In contrast the Permanent can be expressed
as the Hadamard product of two monotone multilinear formulas of
quadratic size.
\end{itemize}

Vikraman Arvind, Pushkar S. Joglekar, and Srikanth Srinivasan. Arithmetic Circuits and the Hadamard Product of Polynomials. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 25-36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2009.2304, author = {Arvind, Vikraman and Joglekar, Pushkar S. and Srinivasan, Srikanth}, title = {{Arithmetic Circuits and the Hadamard Product of Polynomials}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {25--36}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2304}, URN = {urn:nbn:de:0030-drops-23046}, doi = {10.4230/LIPIcs.FSTTCS.2009.2304}, annote = {Keywords: Hadamard product, identity testing, lower bounds, algebraic branching programs} }

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