38 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.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.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
Hardness Magnification near State-Of-The-Art Lower Bounds

Authors: Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam

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


Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.

Cite as

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Copy BibTex To Clipboard

@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27,
  author =	{Oliveira, Igor Carboni and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{Hardness Magnification near State-Of-The-Art Lower Bounds}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{27:1--27:29},
  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.2019.27},
  URN =		{urn:nbn:de:0030-drops-108494},
  doi =		{10.4230/LIPIcs.CCC.2019.27},
  annote =	{Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity}
}
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.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.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
Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Authors: Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki

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


Abstract
Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement.

Cite as

Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki. Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Copy BibTex To Clipboard

@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.45,
  author =	{Golovnev, Alexander and Kulikov, Alexander S. and Smal, Alexander V. and Tamaki, Suguru},
  title =	{{Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{45:1--45:16},
  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.2016.45},
  URN =		{urn:nbn:de:0030-drops-64588},
  doi =		{10.4230/LIPIcs.MFCS.2016.45},
  annote =	{Keywords: circuit complexity, lower bounds, exponential time algorithms, satisfiability}
}
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.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
On the (Non) NP-Hardness of Computing Circuit Complexity

Authors: Cody D. Murray and R. Ryan Williams

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


Abstract
The Minimum Circuit Size Problem (MCSP) is: given the truth table of a Boolean function f and a size parameter k, is the circuit complexity of f at most k? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of its kind, MCSP is not known to be NP-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, MCSP in P would imply there are no pseudorandom functions. Although most NP-complete problems are complete under strong "local" reduction notions such as poly-logarithmic time projections, we show that MCSP is provably not NP-hard under O(n^(1/2-epsilon))-time projections, for every epsilon > 0. We prove that the NP-hardness of MCSP under (logtime-uniform) AC0 reductions would imply extremely strong lower bounds: NP \not\subset P/poly and E \not\subset i.o.-SIZE(2^(delta * n)) for some delta > 0 (hence P = BPP also follows). We show that even the NP-hardness of MCSP under general polynomial-time reductions would separate complexity classes: EXP != NP \cap P/poly, which implies EXP != ZPP. These results help explain why it has been so difficult to prove that MCSP is NP-hard. We also consider the nondeterministic generalization of MCSP: the Nondeterministic Minimum Circuit Size Problem (NMCSP), where one wishes to compute the nondeterministic circuit complexity of a given function. We prove that the Sigma_2 P-hardness of NMCSP, even under arbitrary polynomial-time reductions, would imply EXP \not\subset P/poly.

Cite as

Cody D. Murray and R. Ryan Williams. On the (Non) NP-Hardness of Computing Circuit Complexity. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 365-380, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Copy BibTex To Clipboard

@InProceedings{murray_et_al:LIPIcs.CCC.2015.365,
  author =	{Murray, Cody D. and Williams, R. Ryan},
  title =	{{On the (Non) NP-Hardness of Computing Circuit Complexity}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{365--380},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.365},
  URN =		{urn:nbn:de:0030-drops-50745},
  doi =		{10.4230/LIPIcs.CCC.2015.365},
  annote =	{Keywords: circuit lower bounds, Minimum Circuit Size Problem, NP-completeness, projections, Reductions}
}
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.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
New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs

Authors: Diptarka Chakraborty, A. Pavan, Raghunath Tewari, N. V. Vinodchandran, and Lin Forrest Yang

Published in: LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)


Abstract
We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0. For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved.

Cite as

Diptarka Chakraborty, A. Pavan, Raghunath Tewari, N. V. Vinodchandran, and Lin Forrest Yang. New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 585-595, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2014.585,
  author =	{Chakraborty, Diptarka and Pavan, A. and Tewari, Raghunath and Vinodchandran, N. V. and Yang, Lin Forrest},
  title =	{{New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{585--595},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Raman, Venkatesh and Suresh, S. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.585},
  URN =		{urn:nbn:de:0030-drops-48730},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.585},
  annote =	{Keywords: Reachability, Space complexity, Time-Space Efficient Algorithms, Graphs on Surfaces, Minor Free Graphs, Savitch's Algorithm, BBRS Bound}
}
Document
One Time-traveling Bit is as Good as Logarithmically Many

Authors: Ryan O'Donnell and A. C. Cem Say

Published in: LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)


Abstract
We consider computation in the presence of closed timelike curves (CTCs), as proposed by Deutsch. We focus on the case in which the CTCs carry classical bits (as opposed to qubits). Previously, Aaronson and Watrous showed that computation with polynomially many CTC bits is equivalent in power to PSPACE. On the other hand, Say and Yakaryilmaz showed that computation with just 1 classical CTC bit gives the power of "postselection", thereby upgrading classical randomized computation (BPP) to the complexity class BPP_path and standard quantum computation (BQP) to the complexity class PP. It is natural to ask whether increasing the number of CTC bits from 1 to 2 (or 3, 4, etc.) leads to increased computational power. We show that the answer is no: randomized computation with logarithmically many CTC bits (i.e., polynomially many CTC states) is equivalent to BPP_path. (Similarly, quantum computation augmented with logarithmically many classical CTC bits is equivalent to PP.) Spoilsports with no interest in time travel may view our results as concerning the robustness of the class BPP_path and the computational complexity of sampling from an implicitly defined Markov chain.

Cite as

Ryan O'Donnell and A. C. Cem Say. One Time-traveling Bit is as Good as Logarithmically Many. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 469-480, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{odonnell_et_al:LIPIcs.FSTTCS.2014.469,
  author =	{O'Donnell, Ryan and Cem Say, A. C.},
  title =	{{One Time-traveling Bit is as Good as Logarithmically Many}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{469--480},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Raman, Venkatesh and Suresh, S. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.469},
  URN =		{urn:nbn:de:0030-drops-48646},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.469},
  annote =	{Keywords: Time travel, postselection, Markov chains, randomized computation}
}
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.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.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}
}
  • 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
  • 2 Theory of computation
  • 1 Mathematics of computing → Combinatorial optimization
  • 1 Mathematics of computing → Matroids and greedoids
  • 1 Theory of computation → Computational complexity and cryptography
  • Show More...

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

  • Refine by Type
  • 38 document

  • Refine by Publication Year
  • 8 2008
  • 8 2010
  • 6 2005
  • 3 2015
  • 2 2014
  • 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