14 Search Results for "Hrubes, Pavel"


Document
New Lower Bounds Against Homogeneous Non-Commutative Circuits

Authors: Prerona Chatterjee and Pavel Hrubeš

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


Abstract
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

Cite as

Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.13,
  author =	{Chatterjee, Prerona and Hrube\v{s}, Pavel},
  title =	{{New Lower Bounds Against Homogeneous Non-Commutative Circuits}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{13:1--13:10},
  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.2023.13},
  URN =		{urn:nbn:de:0030-drops-182835},
  doi =		{10.4230/LIPIcs.CCC.2023.13},
  annote =	{Keywords: Algebraic circuit complexity, Non-Commutative Circuits, Homogeneous Computation, Lower bounds against algebraic circuits}
}
Document
Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product

Authors: Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay

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


Abstract
We establish an ε-sensitive hierarchy separation for monotone arithmetic computations. The notion of ε-sensitive monotone lower bounds was recently introduced by Hrubeš [Pavel Hrubeš, 2020]. We show the following: - There exists a monotone polynomial over n variables in VNP that cannot be computed by 2^o(n) size monotone circuits in an ε-sensitive way as long as ε ≥ 2^(-Ω(n)). - There exists a polynomial over n variables that can be computed by polynomial size monotone circuits but cannot be computed by any monotone arithmetic branching program (ABP) of n^o(log n) size, even in an ε-sensitive fashion as long as ε ≥ n^(-Ω(log n)). - There exists a polynomial over n variables that can be computed by polynomial size monotone ABPs but cannot be computed in n^o(log n) size by monotone formulas even in an ε-sensitive way, when ε ≥ n^(-Ω(log n)). - There exists a polynomial over n variables that can be computed by width-4 polynomial size monotone arithmetic branching programs (ABPs) but cannot be computed in 2^o(n^{1/d}) size by monotone, unbounded fan-in formulas of product depth d even in an ε-sensitive way, when ε ≥ 2^(-Ω(n^{1/d})). This yields an ε-sensitive separation of constant-depth monotone formulas and constant-width monotone ABPs. The novel feature of our separations is that in each case the polynomial exhibited is obtained from a graph inner-product polynomial by choosing an appropriate graph topology. The closely related graph inner-product Boolean function for expander graphs was invented by Hayes [Thomas P. Hayes, 2011], also independently by Pitassi [Toniann Pitassi, 2009], in the context of best-partition multiparty communication complexity.

Cite as

Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay. Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 12:1-12:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2022.12,
  author =	{Chattopadhyay, Arkadev and Ghosal, Utsab and Mukhopadhyay, Partha},
  title =	{{Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.12},
  URN =		{urn:nbn:de:0030-drops-174045},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.12},
  annote =	{Keywords: Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations}
}
Document
Monotone Complexity of Spanning Tree Polynomial Re-Visited

Authors: Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having n variables and defined over constant-degree expander graphs, have monotone arithmetic complexity 2^{Ω(n)}. This yields the first strongly exponential lower bound on monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP [S. B. Gashkov and I. S. Sergeev, 2012; Ran Raz and Amir Yehudayoff, 2011; Srikanth Srinivasan, 2020; Bruno Pasqualotto Cavalar et al., 2020; Pavel Hrubeš and Amir Yehudayoff, 2021]. Recently, Hrubeš [Pavel Hrubeš, 2020] initiated a program to prove lower bounds against general arithmetic circuits by proving ε-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for ε ∈ (0,1). The first ε-sensitive lower bound was just proved for a family of polynomials inside VNP by Chattopadhyay, Datta and Mukhopadhyay [Arkadev Chattopadhyay et al., 2021]. We consider the spanning tree polynomial ST_n defined over the complete graph of n vertices and show that the polynomials F_{n-1,n} - ε⋅ ST_{n} and F_{n-1,n} + ε⋅ ST_{n}, defined over (n-1)n variables, have monotone circuit complexity 2^{Ω(n)} if ε ≥ 2^{- Ω(n)} and F_{n-1,n} := ∏_{i = 2}ⁿ (x_{i,1} + ⋯ + x_{i,n}) is the complete set-multilinear polynomial. This provides the first ε-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity. Our two results, thus, are incomparable generalizations of the well known result by Jerrum and Snir [Mark Jerrum and Marc Snir, 1982] which showed that the spanning tree polynomial, defined over complete graphs with n vertices (so the number of variables is (n-1)n), has monotone complexity 2^{Ω(n)}. In particular, the first result is an optimal lower bound and the second result can be thought of as a robust version of the earlier monotone lower bound for the spanning tree polynomial.

Cite as

Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay. Monotone Complexity of Spanning Tree Polynomial Re-Visited. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 39:1-39:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2022.39,
  author =	{Chattopadhyay, Arkadev and Datta, Rajit and Ghosal, Utsab and Mukhopadhyay, Partha},
  title =	{{Monotone Complexity of Spanning Tree Polynomial Re-Visited}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{39:1--39:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.39},
  URN =		{urn:nbn:de:0030-drops-156356},
  doi =		{10.4230/LIPIcs.ITCS.2022.39},
  annote =	{Keywords: Spanning Tree Polynomial, Monotone Computation, Lower Bounds, Communication Complexity}
}
Document
Shadows of Newton Polytopes

Authors: Pavel Hrubeš and Amir Yehudayoff

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


Abstract
We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

Cite as

Pavel Hrubeš and Amir Yehudayoff. Shadows of Newton Polytopes. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 9:1-9:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{hrubes_et_al:LIPIcs.CCC.2021.9,
  author =	{Hrube\v{s}, Pavel and Yehudayoff, Amir},
  title =	{{Shadows of Newton Polytopes}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{9:1--9:23},
  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.2021.9},
  URN =		{urn:nbn:de:0030-drops-142833},
  doi =		{10.4230/LIPIcs.CCC.2021.9},
  annote =	{Keywords: Newton polytope, Monotone arithmetic circuit}
}
Document
Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Authors: Sivaramakrishnan Natarajan Ramamoorthy and Cyrus Rashtchian

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an NP oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the n-dimensional inner product problem with m queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of ω(n/r log m) for r redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time Ω(n^(3/2)/r) for redundancy r ≥ √n in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Koucký, Loff, and Mukhopadhyay.

Cite as

Sivaramakrishnan Natarajan Ramamoorthy and Cyrus Rashtchian. Equivalence of Systematic Linear Data Structures and Matrix Rigidity. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 35:1-35:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{natarajanramamoorthy_et_al:LIPIcs.ITCS.2020.35,
  author =	{Natarajan Ramamoorthy, Sivaramakrishnan and Rashtchian, Cyrus},
  title =	{{Equivalence of Systematic Linear Data Structures and Matrix Rigidity}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.35},
  URN =		{urn:nbn:de:0030-drops-117204},
  doi =		{10.4230/LIPIcs.ITCS.2020.35},
  annote =	{Keywords: matrix rigidity, systematic linear data structures, cell probe model, communication complexity}
}
Document
Invited Talk
From Classical Proof Theory to P versus NP: a Guide to Bounded Theories (Invited Talk)

Authors: Iddo Tzameret

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)


Abstract
This talk explores the question of what does logic and specifically proof theory can tell us about the fundamental hardness questions in computational complexity. We start with a brief description of the main concepts behind bounded arithmetic which is a family of weak formal theories of arithmetic that mirror in a precise manner the world of propositional proofs: if a statement of a given form is provable in a given bounded arithmetic theory then the same statement is suitably translated to a family of propositional formulas with short (polynomial-size) proofs in a corresponding propositional proof system. We will proceed to describe the motivations behind the study of bounded arithmetic theories, their corresponding propositional proof systems, and how they relate to the fundamental complexity class separations and circuit lower bounds questions in computational complexity. We provide a collage of results and recent developments showing how bounded arithmetic and propositional proof complexity form a cohesive framework in which both concrete combinatorial questions about complexity as well as meta-mathematical questions about provability of statements of complexity theory itself, are studied. Specific topics we shall mention are: (i) The bounded reverse mathematics program [Stephen Cook and Phuong Nguyen, 2010]: studying the weakest possible axiomatic assumptions that can prove important results in mathematics and computing (cf. [Iddo Tzameret and Stephen A. Cook, 2017; Pavel Hrubeš and Iddo Tzameret, 2015]), and the correspondence between circuit classes and theories. (ii) The meta-mathematics of computational complexity: what kind of reasoning power do we need in order to prove major results in complexity theory itself, and applications to complexity lower bounds (cf. [Razborov, 1995; Rahul Santhanam and Jan Pich, 2019]). (iii) Proof complexity: the systematic treatment of propositional proofs as combinatorial and algebraic objects and their algorithmic applications (cf. [Samuel Buss, 2012; Tonnian Pitassi and Iddo Tzameret, 2016; Noah Fleming et al., 2019]).

Cite as

Iddo Tzameret. From Classical Proof Theory to P versus NP: a Guide to Bounded Theories (Invited Talk). In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 5:1-5:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{tzameret:LIPIcs.CSL.2020.5,
  author =	{Tzameret, Iddo},
  title =	{{From Classical Proof Theory to P versus NP: a Guide to Bounded Theories}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{5:1--5:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.5},
  URN =		{urn:nbn:de:0030-drops-116482},
  doi =		{10.4230/LIPIcs.CSL.2020.5},
  annote =	{Keywords: Bounded arithmetic, complexity class separations, circuit complexity, proof complexity, weak theories of arithmetic, feasible mathematics}
}
Document
RANDOM
Efficient Black-Box Identity Testing for Free Group Algebras

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)


Abstract
Hrubeš and Wigderson [Pavel Hrubeš and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n.

Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Efficient Black-Box Identity Testing for Free Group Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 57:1-57:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{arvind_et_al:LIPIcs.APPROX-RANDOM.2019.57,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{Efficient Black-Box Identity Testing for Free Group Algebras}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{57:1--57:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.57},
  URN =		{urn:nbn:de:0030-drops-112723},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.57},
  annote =	{Keywords: Rational identity testing, Free group algebra, Noncommutative computation, Randomized algorithms}
}
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)


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@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
Resolution and the Binary Encoding of Combinatorial Principles

Authors: Stefan Dantchev, Nicola Galesi, and Barnaby Martin

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


Abstract
Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(k)} lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [Massimo Lauria et al., 2017] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [Nathan Segerlind et al., 2004], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in Pi_2-form and with no finite model.

Cite as

Stefan Dantchev, Nicola Galesi, and Barnaby Martin. Resolution and the Binary Encoding of Combinatorial Principles. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 6:1-6:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{dantchev_et_al:LIPIcs.CCC.2019.6,
  author =	{Dantchev, Stefan and Galesi, Nicola and Martin, Barnaby},
  title =	{{Resolution and the Binary Encoding of Combinatorial Principles}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{6:1--6:25},
  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.2019.6},
  URN =		{urn:nbn:de:0030-drops-108287},
  doi =		{10.4230/LIPIcs.CCC.2019.6},
  annote =	{Keywords: Proof complexity, k-DNF resolution, binary encodings, Clique and Pigeonhole principle}
}
Document
Track A: Algorithms, Complexity and Games
Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits

Authors: Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)


Abstract
There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n). We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.

Cite as

Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff. Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 72:1-72:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2019.72,
  author =	{Hrube\v{s}, Pavel and Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup and Yehudayoff, Amir},
  title =	{{Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{72:1--72:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.72},
  URN =		{urn:nbn:de:0030-drops-106487},
  doi =		{10.4230/LIPIcs.ICALP.2019.72},
  annote =	{Keywords: Balancing sets, depth-2 threshold circuits, polynomials, majority, weighted thresholds}
}
Document
On Isoperimetric Profiles and Computational Complexity

Authors: Pavel Hrubes and Amir Yehudayoff

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)


Abstract
The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity. We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity. A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.

Cite as

Pavel Hrubes and Amir Yehudayoff. On Isoperimetric Profiles and Computational Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 89:1-89:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2016.89,
  author =	{Hrubes, Pavel and Yehudayoff, Amir},
  title =	{{On Isoperimetric Profiles and Computational Complexity}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{89:1--89:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.89},
  URN =		{urn:nbn:de:0030-drops-61964},
  doi =		{10.4230/LIPIcs.ICALP.2016.89},
  annote =	{Keywords: Monotone computation, separations, communication complexity, isoperimetry}
}
Document
On the Limits of Gate Elimination

Authors: Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov

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


Abstract
Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1/86*n-o(n). All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction.

Cite as

Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov. On the Limits of Gate Elimination. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.46,
  author =	{Golovnev, Alexander and Hirsch, Edward A. and Knop, Alexander and Kulikov, Alexander S.},
  title =	{{On the Limits of Gate Elimination}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{46:1--46:13},
  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.2016.46},
  URN =		{urn:nbn:de:0030-drops-64593},
  doi =		{10.4230/LIPIcs.MFCS.2016.46},
  annote =	{Keywords: circuit complexity, lower bounds, gate elimination}
}
Document
Semantic Versus Syntactic Cutting Planes

Authors: Yuval Filmus, Pavel Hrubeš, and Massimo Lauria

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)


Abstract
In this paper, we compare the strength of the semantic and syntactic version of the cutting planes proof system. First, we show that the lower bound technique of Pudlák applies also to semantic cutting planes: the proof system has feasible interpolation via monotone real circuits, which gives an exponential lower bound on lengths of semantic cutting planes refutations. Second, we show that semantic refutations are stronger than syntactic ones. In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes. In other words, syntactic cutting planes does not p-simulate semantic cutting planes. We also give two incompatible integer inequalities which require exponential length refutation in syntactic cutting planes. Finally, we pose the following problem, which arises in connection with semantic inference of arity larger than two: can every multivariate non-decreasing real function be expressed as a composition of non-decreasing real functions in two variables?

Cite as

Yuval Filmus, Pavel Hrubeš, and Massimo Lauria. Semantic Versus Syntactic Cutting Planes. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 35:1-35:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{filmus_et_al:LIPIcs.STACS.2016.35,
  author =	{Filmus, Yuval and Hrube\v{s}, Pavel and Lauria, Massimo},
  title =	{{Semantic Versus Syntactic Cutting Planes}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{35:1--35:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.35},
  URN =		{urn:nbn:de:0030-drops-57367},
  doi =		{10.4230/LIPIcs.STACS.2016.35},
  annote =	{Keywords: proof complexity, cutting planes, lower bounds}
}
Document
Circuits with Medium Fan-In

Authors: Pavel Hrubes and Anup Rao

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


Abstract
We consider boolean circuits in which every gate may compute an arbitrary boolean function of k other gates, for a parameter k. We give an explicit function $f:{0,1}^n -> {0,1} that requires at least Omega(log^2(n)) non-input gates when k = 2n/3. When the circuit is restricted to being layered and depth 2, we prove a lower bound of n^(Omega(1)) on the number of non-input gates. When the circuit is a formula with gates of fan-in k, we give a lower bound Omega(n^2/k*log(n)) on the total number of gates. Our model is connected to some well known approaches to proving lower bounds in complexity theory. Optimal lower bounds for the Number-On-Forehead model in communication complexity, or for bounded depth circuits in AC_0, or extractors for varieties over small fields would imply strong lower bounds in our model. On the other hand, new lower bounds for our model would prove new time-space tradeoffs for branching programs and impossibility results for (fan-in 2) circuits with linear size and logarithmic depth. In particular, our lower bound gives a different proof for a known time-space tradeoff for oblivious branching programs.

Cite as

Pavel Hrubes and Anup Rao. Circuits with Medium Fan-In. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hrubes_et_al:LIPIcs.CCC.2015.381,
  author =	{Hrubes, Pavel and Rao, Anup},
  title =	{{Circuits with Medium Fan-In}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{381--391},
  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.2015.381},
  URN =		{urn:nbn:de:0030-drops-50528},
  doi =		{10.4230/LIPIcs.CCC.2015.381},
  annote =	{Keywords: Boolean circuit, Complexity, Communication Complexity}
}
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