26 Search Results for "Agrawal, Manindra"

Document
DOI: 10.4230/LIPIcs.CCC.2024.14
Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

Authors: Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.
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Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14,
  author =	{B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
  title =	{{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{14:1--14:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14},
  URN =		{urn:nbn:de:0030-drops-204100},
  doi =		{10.4230/LIPIcs.CCC.2024.14},
  annote =	{Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.20
Lower Bounds for Set-Multilinear Branching Programs

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

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.
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Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20,
  author =	{Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir},
  title =	{{Lower Bounds for Set-Multilinear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20},
  URN =		{urn:nbn:de:0030-drops-204167},
  doi =		{10.4230/LIPIcs.CCC.2024.20},
  annote =	{Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.17
Barcode Selection and Layout Optimization in Spatial Transcriptomics

Authors: Frederik L. Jatzkowski, Antonia Schmidt, Robert Mank, Steffen Schüler, and Matthias Müller-Hannemann

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
An important special case of the quadratic assignment problem arises in the synthesis of DNA microarrays for high-resolution spatial transcriptomics. The task is to select a suitable subset from a set of barcodes, i. e. short DNA strings that serve as unique identifiers, and to assign the selected barcodes to positions on a two-dimensional array in such a way that a position-dependent cost function is minimized. A typical microarray with dimensions of 768×1024 requires 786,432 many barcodes to be placed, leading to very challenging large-scale combinatorial optimization problems. The general quadratic assignment problem is well-known for its hardness, both in theory and in practice. It turns out that this also holds for the special case of the barcode layout problem. We show that the problem is even hard to approximate: It is MaxSNP-hard. An ILP formulation theoretically allows the computation of optimal results, but it is only applicable for tiny instances. Therefore, we have developed layout constructing and improving heuristics with the aim of computing near-optimal solutions for instances of realistic size. These include a sorting-based algorithm, a greedy algorithm, 2-OPT-based local search and a genetic algorithm. To assess the quality of the results, we compare the generated solutions with the expected cost of a random layout and with lower bounds. A combination of the greedy algorithm and 2-OPT local search produces the most promising results in terms of both quality and runtime. Solutions to large-scale instances with arrays of dimension 768×1024 show a 37% reduction in cost over a random solution and can be computed in about 3 minutes. Since the universe of suitable barcodes is much larger than the number of barcodes needed, this can be exploited. Experiments with different surpluses of barcodes show that a significant improvement in layout quality can be achieved at the cost of a reasonable increase in runtime. Another interesting finding is that the restriction of the barcode design space by biochemical constraints is actually beneficial for the overall layout cost.
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Frederik L. Jatzkowski, Antonia Schmidt, Robert Mank, Steffen Schüler, and Matthias Müller-Hannemann. Barcode Selection and Layout Optimization in Spatial Transcriptomics. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jatzkowski_et_al:LIPIcs.SEA.2024.17,
  author =	{Jatzkowski, Frederik L. and Schmidt, Antonia and Mank, Robert and Sch\"{u}ler, Steffen and M\"{u}ller-Hannemann, Matthias},
  title =	{{Barcode Selection and Layout Optimization in Spatial Transcriptomics}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{17:1--17:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.17},
  URN =		{urn:nbn:de:0030-drops-203821},
  doi =		{10.4230/LIPIcs.SEA.2024.17},
  annote =	{Keywords: Spatial Transcriptomics, Array Layout, Optimization, Computational Complexity, GPU Computing, Integer Linear Programming, Metaheuristics}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.16
NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors: Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.
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Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baraskar_et_al:LIPIcs.ICALP.2024.16,
  author =	{Baraskar, Omkar and Dewan, Agrim and Saha, Chandan and Sinha, Pulkit},
  title =	{{NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.16},
  URN =		{urn:nbn:de:0030-drops-201598},
  doi =		{10.4230/LIPIcs.ICALP.2024.16},
  annote =	{Keywords: Equivalence testing, MCSP, sparse polynomials, 3SAT}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.23
A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization

Authors: Pranjal Dutta, Nitin Saxena, and Thomas Thierauf

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

Abstract
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.
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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)

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@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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.11
Constructing Faithful Homomorphisms over Fields of Finite Characteristic

Authors: Prerona Chatterjee and Ramprasad Saptharishi

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

Abstract
We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting.
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Prerona Chatterjee and Ramprasad Saptharishi. Constructing Faithful Homomorphisms over Fields of Finite Characteristic. 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. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2019.11,
  author =	{Chatterjee, Prerona and Saptharishi, Ramprasad},
  title =	{{Constructing Faithful Homomorphisms over Fields of Finite Characteristic}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{11:1--11:14},
  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.2019.11},
  URN =		{urn:nbn:de:0030-drops-115733},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.11},
  annote =	{Keywords: Faithful Homomorphisms, Identity Testing, Algebraic Independence, Finite characteristic fields}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.6
Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

Authors: Ramprasad Saptharishi and Anamay Tengse

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

Abstract
We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions: - An explicit hitting set of quasipolynomial size for UPT circuits, - An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), - An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016].
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Ramprasad Saptharishi and Anamay Tengse. Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees. 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. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{saptharishi_et_al:LIPIcs.FSTTCS.2018.6,
  author =	{Saptharishi, Ramprasad and Tengse, Anamay},
  title =	{{Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{6:1--6:19},
  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.2018.6},
  URN =		{urn:nbn:de:0030-drops-99050},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.6},
  annote =	{Keywords: Unambiguous Circuits, Read-once Oblivious ABPs, Polynomial Identity Testing, Lower Bounds, Algebraic Circuit Complexity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.6
Integer Factoring Using Small Algebraic Dependencies

Authors: Manindra Agrawal, Nitin Saxena, and Shubham Sahai Srivastava

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring. In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques.
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Manindra Agrawal, Nitin Saxena, and Shubham Sahai Srivastava. Integer Factoring Using Small Algebraic Dependencies. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{agrawal_et_al:LIPIcs.MFCS.2016.6,
  author =	{Agrawal, Manindra and Saxena, Nitin and Srivastava, Shubham Sahai},
  title =	{{Integer Factoring Using Small Algebraic Dependencies}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.6},
  URN =		{urn:nbn:de:0030-drops-64234},
  doi =		{10.4230/LIPIcs.MFCS.2016.6},
  annote =	{Keywords: integer, factorization, factoring integers, algebraic dependence, dependencies}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2015.221
Dimension, Pseudorandomness and Extraction of Pseudorandomness

Authors: Manindra Agrawal, Diptarka Chakraborty, Debarati Das, and Satyadev Nandakumar

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Abstract
In this paper we propose a quantification of distributions on a set of strings, in terms of how close to pseudorandom a distribution is. The quantification is an adaptation of the theory of dimension of sets of infinite sequences introduced by Lutz. Adapting Hitchcock's work, we also show that the logarithmic loss incurred by a predictor on a distribution is quantitatively equivalent to the notion of dimension we define. Roughly, this captures the equivalence between pseudorandomness defined via indistinguishability and via unpredictability. Later we show some natural properties of our notion of dimension. We also do a comparative study among our proposed notion of dimension and two well known notions of computational analogue of entropy, namely HILL-type pseudo min-entropy and next-bit pseudo Shannon entropy. Further, we apply our quantification to the following problem. If we know that the dimension of a distribution on the set of n-length strings is s in (0,1], can we extract out O(sn) pseudorandom bits out of the distribution? We show that to construct such extractor, one need at least Omega(log n) bits of pure randomness. However, it is still open to do the same using O(log n) random bits. We show that deterministic extraction is possible in a special case - analogous to the bit-fixing sources introduced by Chor et al., which we term nonpseudorandom bit-fixing source. We adapt the techniques of Gabizon, Raz and Shaltiel to construct a deterministic pseudorandom extractor for this source. By the end, we make a little progress towards P vs. BPP problem by showing that existence of optimal stretching function that stretches O(log n) input bits to produce n output bits such that output distribution has dimension s in (0,1], implies P=BPP.
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Manindra Agrawal, Diptarka Chakraborty, Debarati Das, and Satyadev Nandakumar. Dimension, Pseudorandomness and Extraction of Pseudorandomness. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 221-235, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2015.221,
  author =	{Agrawal, Manindra and Chakraborty, Diptarka and Das, Debarati and Nandakumar, Satyadev},
  title =	{{Dimension, Pseudorandomness and Extraction of Pseudorandomness}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{221--235},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.221},
  URN =		{urn:nbn:de:0030-drops-56184},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.221},
  annote =	{Keywords: Pseudorandomness, Dimension, Martingale, Unpredictability, Pseudoentropy, Pseudorandom Extractor, Hard Function}
}
Document
DOI: 10.4230/DagRep.4.9.85
Algebra in Computational Complexity (Dagstuhl Seminar 14391)

Authors: Manindra Agrawal, Valentine Kabanets, Thomas Thierauf, and Christopher Umans

Published in: Dagstuhl Reports, Volume 4, Issue 9 (2015)

Abstract
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress.
Cite as

Manindra Agrawal, Valentine Kabanets, Thomas Thierauf, and Christopher Umans. Algebra in Computational Complexity (Dagstuhl Seminar 14391). In Dagstuhl Reports, Volume 4, Issue 9, pp. 85-105, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{agrawal_et_al:DagRep.4.9.85,
  author =	{Agrawal, Manindra and Kabanets, Valentine and Thierauf, Thomas and Umans, Christopher},
  title =	{{Algebra in Computational Complexity (Dagstuhl Seminar 14391)}},
  pages =	{85--105},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2015},
  volume =	{4},
  number =	{9},
  editor =	{Agrawal, Manindra and Kabanets, Valentine and Thierauf, Thomas and Umans, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.9.85},
  URN =		{urn:nbn:de:0030-drops-48851},
  doi =		{10.4230/DagRep.4.9.85},
  annote =	{Keywords: Computational Complexity, lower bounds, approximazation, pseudo-randomness, derandomization, circuits}
}
Document
DOI: 10.4230/DagRep.2.10.60
Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

Authors: Manindra Agrawal, Thomas Thierauf, and Christopher Umans

Published in: Dagstuhl Reports, Volume 2, Issue 10 (2013)

Abstract
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings.
Cite as

Manindra Agrawal, Thomas Thierauf, and Christopher Umans. Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421). In Dagstuhl Reports, Volume 2, Issue 10, pp. 60-78, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{agrawal_et_al:DagRep.2.10.60,
  author =	{Agrawal, Manindra and Thierauf, Thomas and Umans, Christopher},
  title =	{{Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)}},
  pages =	{60--78},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{2},
  number =	{10},
  editor =	{Agrawal, Manindra and Thierauf, Thomas and Umans, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.10.60},
  URN =		{urn:nbn:de:0030-drops-39034},
  doi =		{10.4230/DagRep.2.10.60},
  annote =	{Keywords: Computational Complexity, lower bounds, approximazation, pseudo-randomness, derandomization, circuits}
}
Document
DOI: 10.4230/DagSemProc.09421.1
09421 Abstracts Collection – Algebraic Methods in Computational Complexity

Authors: Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract
From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
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Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans. 09421 Abstracts Collection – Algebraic Methods in Computational Complexity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{agrawal_et_al:DagSemProc.09421.1,
  author =	{Agrawal, Manindra and Fortnow, Lance and Thierauf, Thomas and Umans, Christopher},
  title =	{{09421 Abstracts Collection – Algebraic Methods in Computational Complexity}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--22},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.1},
  URN =		{urn:nbn:de:0030-drops-24181},
  doi =		{10.4230/DagSemProc.09421.1},
  annote =	{Keywords: Computational Complexity, Algebra}
}
Document
DOI: 10.4230/DagSemProc.09421.2
09421 Executive Summary – Algebraic Methods in Computational Complexity

Authors: Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract
The seminar brought together more than 50 researchers covering a wide spectrum of complexity theory. The focus on algebraic methods showed once again the great importance of algebraic techniques for theoretical computer science. We had almost 30 talks, most of them about 40 minutes leaving ample room for discussions. We also had a much appreciated open problem session. The talks ranged over a broad assortment of subjects with the underlying theme of using algebraic techniques. It was very fruitful and has hopefully initiated new directions in research. Several participants specifically mentioned that they appreciated the particular focus on a common class of techniques (rather than end results) as a unifying theme of the workshop. We look forward to our next meeting!
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Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans. 09421 Executive Summary – Algebraic Methods in Computational Complexity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{agrawal_et_al:DagSemProc.09421.2,
  author =	{Agrawal, Manindra and Fortnow, Lance and Thierauf, Thomas and Umans, Christopher},
  title =	{{09421 Executive Summary – Algebraic Methods in Computational Complexity}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.2},
  URN =		{urn:nbn:de:0030-drops-24100},
  doi =		{10.4230/DagSemProc.09421.2},
  annote =	{Keywords: Computational Complexity, Algebra}
}
Document
DOI: 10.4230/DagSemProc.09421.3
An Axiomatic Approach to Algebrization

Authors: Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract
Non-relativization of complexity issues can be interpreted as giving some evidence that these issues cannot be resolved by "black-box" techniques. In the early 1990's, a sequence of important non-relativizing results was proved, mainly using algebraic techniques. Two approaches have been proposed to understand the power and limitations of these algebraic techniques: (1) Fortnow gives a construction of a class of oracles which have a similar algebraic and logical structure, although they are arbitrarily powerful. He shows that many of the non-relativizing results proved using algebraic techniques hold for all such oracles, but he does not show, e.g., that the outcome of the "P vs. NP" question differs between different oracles in that class. (2) Aaronson and Wigderson give definitions of algebrizing separations and collapses of complexity classes, by comparing classes relative to one oracle to classes relative to an algebraic extension of that oracle. Using these definitions, they show both that the standard collapses and separations "algebrize" and that many of the open questions in complexity fail to "algebrize", suggesting that the arithmetization technique is close to its limits. However, it is unclear how to formalize algebrization of more complicated complexity statements than collapses or separations, and whether the algebrizing statements are, e.g., closed under modus ponens; so it is conceivable that several algebrizing premises could imply (in a relativizing way) a non-algebrizing conclusion. Here, building on the work of Arora, Impagliazzo, and Vazirani [4], we propose an axiomatic approach to "algebrization", which complements and clarifies the approaches of Fortnow and Aaronso&Wigderson. We present logical theories formalizing the notion of algebrizing techniques so that most algebrizing results are provable within our theories and separations requiring non-algebrizing techniques are independent of them. Our theories extend the [AIV] theory formalizing relativization by adding an Arithmetic Checkability axiom. We show the following: (i) Arithmetic checkability holds relative to arbitrarily powerful oracles (since Fortnow's algebraic oracles all satisfy Arithmetic Checkability axiom); by contrast, Local Checkability of [AIV] restricts the oracle power to NP cap co-NP. (ii) Most of the algebrizing collapses and separations from [AW], such as IP = PSPACE, NP subset ZKIP if one-way functions exist, MA-EXP not in P/poly, etc., are provable from Arithmetic Checkability. (iii) Many of the open complexity questions (shown to require nonalgebrizing techniques in [AW]), such as "P vs. NP", "NP vs. BPP", etc., cannot be proved from Arithmetic Checkability. (iv) Arithmetic Checkability is also insufficient to prove one known result, NEXP = MIP.
Cite as

Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. An Axiomatic Approach to Algebrization. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{impagliazzo_et_al:DagSemProc.09421.3,
  author =	{Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina},
  title =	{{An Axiomatic Approach to Algebrization}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--19},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.3},
  URN =		{urn:nbn:de:0030-drops-24150},
  doi =		{10.4230/DagSemProc.09421.3},
  annote =	{Keywords: Oracles, arithmetization, algebrization}
}
Document
DOI: 10.4230/DagSemProc.09421.4
Deterministic approximation algorithms for the nearest codeword problem

Authors: Noga Alon, Rina Panigrahy, and Sergey Yekhanin

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract
The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c; and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k / k) for all k < n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.
Cite as

Noga Alon, Rina Panigrahy, and Sergey Yekhanin. Deterministic approximation algorithms for the nearest codeword problem. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{alon_et_al:DagSemProc.09421.4,
  author =	{Alon, Noga and Panigrahy, Rina and Yekhanin, Sergey},
  title =	{{Deterministic approximation algorithms for the nearest codeword problem}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--13},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.4},
  URN =		{urn:nbn:de:0030-drops-24133},
  doi =		{10.4230/DagSemProc.09421.4},
  annote =	{Keywords: }
}
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