33 Search Results for "Thierauf, Thomas"


Document
Border Complexity of Symbolic Determinant Under Rank One Restriction

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

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


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

Cite as

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


Copy BibTex To Clipboard

@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.2,
  author =	{Chatterjee, Abhranil and Ghosh, Sumanta and Gurjar, Rohit and Raj, Roshan},
  title =	{{Border Complexity of Symbolic Determinant Under Rank One Restriction}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{2:1--2:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.2},
  URN =		{urn:nbn:de:0030-drops-182721},
  doi =		{10.4230/LIPIcs.CCC.2023.2},
  annote =	{Keywords: Border Complexity, Symbolic Determinant, Valuated Matroid}
}
Document
Arithmetic Circuit Complexity of Division and Truncation

Authors: Pranjal Dutta, Gorav Jindal, Anurag Pandey, and Amit Sinhababu

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


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

Cite as

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


Copy BibTex To Clipboard

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

Cite as

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


Copy BibTex To Clipboard

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

Authors: Amit Sinhababu and Thomas Thierauf

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


Abstract
Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly(s^(log s)).

Cite as

Amit Sinhababu and Thomas Thierauf. Factorization of Polynomials Given By Arithmetic Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Copy BibTex To Clipboard

@InProceedings{sinhababu_et_al:LIPIcs.CCC.2020.33,
  author =	{Sinhababu, Amit and Thierauf, Thomas},
  title =	{{Factorization of Polynomials Given By Arithmetic Branching Programs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{33:1--33:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.33},
  URN =		{urn:nbn:de:0030-drops-125854},
  doi =		{10.4230/LIPIcs.CCC.2020.33},
  annote =	{Keywords: Arithmetic Branching Program, Multivariate Polynomial Factorization, Hensel Lifting, Newton Iteration, Hardness vs Randomness}
}
Document
Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

Authors: Rohit Gurjar, Thomas Thierauf, and Nisheeth K. Vishnoi

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


Abstract
We present a geometric approach towards derandomizing the {Isolation Lemma} by Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a quasi-polynomial family of weights, where each weight is an integer and quasi-polynomially bounded, that can isolate a vertex in any 0/1 polytope for which each face lies in an affine space defined by a totally unimodular matrix. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the Isolation Lemma for {bipartite perfect matching} and {matroid intersection}. We prove our result by associating a {lattice} to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of vectors of length within 3/2 of the shortest vector in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of circuits of size within 3/2 of the shortest circuit in a regular matroid. This is the technical core of the paper and relies on a variant of Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids.

Cite as

Rohit Gurjar, Thomas Thierauf, and Nisheeth K. Vishnoi. Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Copy BibTex To Clipboard

@InProceedings{gurjar_et_al:LIPIcs.ICALP.2018.74,
  author =	{Gurjar, Rohit and Thierauf, Thomas and Vishnoi, Nisheeth K.},
  title =	{{Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{74:1--74:14},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.74},
  URN =		{urn:nbn:de:0030-drops-90782},
  doi =		{10.4230/LIPIcs.ICALP.2018.74},
  annote =	{Keywords: Derandomization, Isolation Lemma, Total unimodularity, Near-shortest vectors in Lattices, Regular matroids}
}
Document
Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

Authors: Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, and Christopher Umans

Published in: Dagstuhl Reports, Volume 6, Issue 10 (2017)


Abstract
Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue 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 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). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress.

Cite as

Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, and Christopher Umans. Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411). In Dagstuhl Reports, Volume 6, Issue 10, pp. 13-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Copy BibTex To Clipboard

@Article{kabanets_et_al:DagRep.6.10.13,
  author =	{Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher},
  title =	{{Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)}},
  pages =	{13--32},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{6},
  number =	{10},
  editor =	{Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.6.10.13},
  URN =		{urn:nbn:de:0030-drops-69504},
  doi =		{10.4230/DagRep.6.10.13},
  annote =	{Keywords: Computational Complexity, lower bounds, approximation, pseudo-randomness, derandomization, circuits}
}
Document
Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs

Authors: Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity n^(O(log(n))). In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

Cite as

Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 323-346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Copy BibTex To Clipboard

@InProceedings{gurjar_et_al:LIPIcs.CCC.2015.323,
  author =	{Gurjar, Rohit and Korwar, Arpita and Saxena, Nitin and Thierauf, Thomas},
  title =	{{Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{323--346},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.323},
  URN =		{urn:nbn:de:0030-drops-50647},
  doi =		{10.4230/LIPIcs.CCC.2015.323},
  annote =	{Keywords: PIT, Hitting-set, Sum of ROABPs, Evaluation Dimension, Rank Concentration}
}
Document
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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.

Cite as

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)


Copy BibTex To Clipboard

@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-dev.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
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!

Cite as

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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.4},
  URN =		{urn:nbn:de:0030-drops-24133},
  doi =		{10.4230/DagSemProc.09421.4},
  annote =	{Keywords: }
}
Document
Learning Parities in the Mistake-Bound model

Authors: Harry Buhrman, David Garcia-Soriano, and Arie Matsliah

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


Abstract
We study the problem of learning parity functions that depend on at most $k$ variables ($k$-parities) attribute-efficiently in the mistake-bound model. We design a simple, deterministic, polynomial-time algorithm for learning $k$-parities with mistake bound $O(n^{1-frac{c}{k}})$, for any constant $c > 0$. This is the first polynomial-time algorithms that learns $omega(1)$-parities in the mistake-bound model with mistake bound $o(n)$. Using the standard conversion techniques from the mistake-bound model to the PAC model, our algorithm can also be used for learning $k$-parities in the PAC model. In particular, this implies a slight improvement on the results of Klivans and Servedio cite{rocco} for learning $k$-parities in the PAC model. We also show that the $widetilde{O}(n^{k/2})$ time algorithm from cite{rocco} that PAC-learns $k$-parities with optimal sample complexity can be extended to the mistake-bound model.

Cite as

Harry Buhrman, David Garcia-Soriano, and Arie Matsliah. Learning Parities in the Mistake-Bound model. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


Copy BibTex To Clipboard

@InProceedings{buhrman_et_al:DagSemProc.09421.5,
  author =	{Buhrman, Harry and Garcia-Soriano, David and Matsliah, Arie},
  title =	{{Learning Parities in the Mistake-Bound model}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--9},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.5},
  URN =		{urn:nbn:de:0030-drops-24178},
  doi =		{10.4230/DagSemProc.09421.5},
  annote =	{Keywords: Attribute-efficient learning, parities, mistake-bound}
}
Document
Planar Graph Isomorphism is in Log-Space

Authors: Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner

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


Abstract
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph.

Cite as

Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. Planar Graph Isomorphism is in Log-Space. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


Copy BibTex To Clipboard

@InProceedings{datta_et_al:DagSemProc.09421.6,
  author =	{Datta, Samir and Limaye, Nutan and Nimbhorkar, Prajakta and Thierauf, Thomas and Wagner, Fabian},
  title =	{{Planar Graph Isomorphism is in Log-Space}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--32},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.6},
  URN =		{urn:nbn:de:0030-drops-24169},
  doi =		{10.4230/DagSemProc.09421.6},
  annote =	{Keywords: Planar Graphs, Graph Isomorphism, Logspace}
}
  • Refine by Author
  • 16 Thierauf, Thomas
  • 7 Buhrman, Harry
  • 7 Fortnow, Lance
  • 6 Agrawal, Manindra
  • 5 Umans, Christopher
  • Show More...

  • Refine by Classification
  • 3 Theory of computation → Algebraic complexity theory
  • 1 Mathematics of computing → Combinatorial optimization
  • 1 Mathematics of computing → Matroids and greedoids
  • 1 Theory of computation
  • 1 Theory of computation → Computational complexity and cryptography
  • Show More...

  • Refine by Keyword
  • 5 Computational Complexity
  • 5 lower bounds
  • 4 derandomization
  • 3 Computational complexity
  • 3 circuits
  • Show More...

  • Refine by Type
  • 33 document

  • Refine by Publication Year
  • 8 2008
  • 8 2010
  • 6 2005
  • 2 2015
  • 2 2021
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail