16 Search Results for "Håstad, Johan"


Document
APPROX
The Biased Homogeneous r-Lin Problem

Authors: Suprovat Ghoshal

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)


Abstract
The p-biased Homogeneous r-Lin problem (Hom-r-Lin_p) is the following: given a homogeneous system of r-variable equations over m{F}₂, the goal is to find an assignment of relative weight p that satisfies the maximum number of equations. In a celebrated work, Håstad (JACM 2001) showed that the unconstrained variant of this i.e., Max-3-Lin, is hard to approximate beyond a factor of 1/2. This is also tight due to the naive random guessing algorithm which sets every variable uniformly from {0,1}. Subsequently, Holmerin and Khot (STOC 2004) showed that the same holds for the balanced Hom-r-Lin problem as well. In this work, we explore the approximability of the Hom-r-Lin_p problem beyond the balanced setting (i.e., p ≠ 1/2), and investigate whether the (p-biased) random guessing algorithm is optimal for every p. Our results include the following: - The Hom-r-Lin_p problem has no efficient 1/2 + 1/2 (1 - 2p)^{r-2} + ε-approximation algorithm for every p if r is even, and for p ∈ (0,1/2] if r is odd, unless NP ⊂ ∪_{ε>0}DTIME(2^{n^ε}). - For any r and any p, there exists an efficient 1/2 (1 - e^{-2})-approximation algorithm for Hom-r-Lin_p. We show that this is also tight for odd values of r (up to o_r(1)-additive factors) assuming the Unique Games Conjecture. Our results imply that when r is even, then for large values of r, random guessing is near optimal for every p. On the other hand, when r is odd, our results illustrate an interesting contrast between the regimes p ∈ (0,1/2) (where random guessing is near optimal) and p → 1 (where random guessing is far from optimal). A key technical contribution of our work is a generalization of Håstad’s 3-query dictatorship test to the p-biased setting.

Cite as

Suprovat Ghoshal. The Biased Homogeneous r-Lin Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ghoshal:LIPIcs.APPROX/RANDOM.2022.47,
  author =	{Ghoshal, Suprovat},
  title =	{{The Biased Homogeneous r-Lin Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.47},
  URN =		{urn:nbn:de:0030-drops-171695},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.47},
  annote =	{Keywords: Biased Approximation Resistance, Constraint Satisfaction Problems}
}
Document
APPROX
Some Results on Approximability of Minimum Sum Vertex Cover

Authors: Aleksa Stanković

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)


Abstract
We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering which minimizes the sum of the cover times over all edges in the graph. In this work we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.014. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work of Bansal, Batra, Farhadi, and Tetali. We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali, and show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.

Cite as

Aleksa Stanković. Some Results on Approximability of Minimum Sum Vertex Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{stankovic:LIPIcs.APPROX/RANDOM.2022.50,
  author =	{Stankovi\'{c}, Aleksa},
  title =	{{Some Results on Approximability of Minimum Sum Vertex Cover}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{50:1--50:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.50},
  URN =		{urn:nbn:de:0030-drops-171722},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.50},
  annote =	{Keywords: Hardness of approximation, approximability, approximation algorithms, Label Cover, Unique Games Conjecture, Vertex Cover}
}
Document
Improved Pseudorandom Generators for AC⁰ Circuits

Authors: Xin Lyu

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


Abstract
We give PRG for depth-d, size-m AC⁰ circuits with seed length O(log^{d-1}(m)log(m/ε)log log(m)). Our PRG improves on previous work [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] from various aspects. It has optimal dependence on 1/ε and is only one "log log(m)" away from the lower bound barrier. For the case of d = 2, the seed length tightly matches the best-known PRG for CNFs [Anindya De et al., 2010; Avishay Tal, 2017]. There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for AC⁰. Previous works [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] usually built PRGs on the Ajtai-Wigderson framework [Miklós Ajtai and Avi Wigderson, 1989]. Compared with them, the partitioning approach avoids the extra "log(n)" factor that usually arises from the Ajtai-Wigderson framework, allowing us to get the almost-tight seed length. The partitioning approach is quite general, and we believe it can help design PRGs for classes beyond constant-depth circuits. Second, improving and extending [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021], we prove a full derandomization of the powerful multi-switching lemma [Johan Håstad, 2014]. We show that one can use a short random seed to sample a restriction, such that a family of DNFs simultaneously simplifies under the restriction with high probability. This answers an open question in [Zander Kelley, 2021]. Previous derandomizations were either partial (that is, they pseudorandomly choose variables to restrict, and then fix those variables to truly-random bits) or had sub-optimal seed length. In our application, having a fully-derandomized switching lemma is crucial, and the randomness-efficiency of our derandomization allows us to get an almost-tight seed length.

Cite as

Xin Lyu. Improved Pseudorandom Generators for AC⁰ Circuits. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 34:1-34:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lyu:LIPIcs.CCC.2022.34,
  author =	{Lyu, Xin},
  title =	{{Improved Pseudorandom Generators for AC⁰ Circuits}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{34:1--34:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.34},
  URN =		{urn:nbn:de:0030-drops-165963},
  doi =		{10.4230/LIPIcs.CCC.2022.34},
  annote =	{Keywords: pseudorandom generators, derandomization, switching Lemmas, AC⁰}
}
Document
APPROX
Revisiting Alphabet Reduction in Dinur’s PCP

Authors: Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.

Cite as

Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep. Revisiting Alphabet Reduction in Dinur’s PCP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2020.34,
  author =	{Guruswami, Venkatesan and Opr\v{s}al, Jakub and Sandeep, Sai},
  title =	{{Revisiting Alphabet Reduction in Dinur’s PCP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.34},
  URN =		{urn:nbn:de:0030-drops-126372},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.34},
  annote =	{Keywords: PCP theorem, CSP, discrete Fourier analysis, label cover, long code}
}
Document
Limits of Preprocessing

Authors: Yuval Filmus, Yuval Ishai, Avi Kaplan, and Guy Kindler

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


Abstract
It is a classical result that the inner product function cannot be computed by an AC⁰ circuit [Merrick L. Furst et al., 1981; Miklós Ajtai, 1983; Johan Håstad, 1986]. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output n + n/(log^{ω(1)} n) bits. Our methods extend to many other functions, including pseudorandom functions, and imply a (weak but nontrivial) limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the main conjecture with the question of learning AC⁰ under simple input distributions.

Cite as

Yuval Filmus, Yuval Ishai, Avi Kaplan, and Guy Kindler. Limits of Preprocessing. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 17:1-17:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{filmus_et_al:LIPIcs.CCC.2020.17,
  author =	{Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Kindler, Guy},
  title =	{{Limits of Preprocessing}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{17:1--17:22},
  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.2020.17},
  URN =		{urn:nbn:de:0030-drops-125697},
  doi =		{10.4230/LIPIcs.CCC.2020.17},
  annote =	{Keywords: circuit, communication complexity, IPPP, preprocessing, PRF, simultaneous messages}
}
Document
Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

Authors: Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov

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


Abstract
We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

Cite as

Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28,
  author =	{de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry},
  title =	{{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{28:1--28:24},
  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.2020.28},
  URN =		{urn:nbn:de:0030-drops-125804},
  doi =		{10.4230/LIPIcs.CCC.2020.28},
  annote =	{Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs}
}
Document
Stable Memoryless Queuing under Contention

Authors: Paweł Garncarek, Tomasz Jurdziński, and Dariusz R. Kowalski

Published in: LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)


Abstract
In this work we study stability of local memoryless packet scheduling policies in a distributed system of n nodes/queues under contention. The local policies at nodes may only access their current local queues, and have no other feedback from the underlying distributed system. Moreover, their memory is limited to some basic parameters. The packets arrive at queues according to arrival patterns controlled by an adversary restricted only by injection rate rho and burstiness b, or driven by a stochastic process; the former model analyzes worst-case stability while the latter - average case. We assume that the underlying distributed system is a classic shared channel, in which no two packets could be successfully scheduled (and removed from queues) at the same time. We show that there is a local memoryless scheduling policy which is both adversarially and stochastically stable for injection rates Omega(1/log n). Another algorithm achieves even higher - constant - stable injection rate, but only for a bounded range of burstiness. The first algorithm is utilizing properties of interleaved ultra-selectors, for which we prove stronger properties than known so far, while the second one is based on entirely new concept of selector with thresholds, unlike previously considered binary selectors/codes in the literature. Note that popular Backoff algorithms, some of which achieve stability for constant (stochastic) injection rates [Johan Håstad et al., 1996], use memory to record current state (e.g., the number of unsuccessful transmissions or the result of random sampling in each window) as well as randomization and feedback from the channel; unlike solutions in this work, which are memoryless and do not rely on randomization or channel feedback (thus, could be used independently from the link layer protocols). {}

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Paweł Garncarek, Tomasz Jurdziński, and Dariusz R. Kowalski. Stable Memoryless Queuing under Contention. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{garncarek_et_al:LIPIcs.DISC.2019.17,
  author =	{Garncarek, Pawe{\l} and Jurdzi\'{n}ski, Tomasz and Kowalski, Dariusz R.},
  title =	{{Stable Memoryless Queuing under Contention}},
  booktitle =	{33rd International Symposium on Distributed Computing (DISC 2019)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-126-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{146},
  editor =	{Suomela, Jukka},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.17},
  URN =		{urn:nbn:de:0030-drops-113244},
  doi =		{10.4230/LIPIcs.DISC.2019.17},
  annote =	{Keywords: packet scheduling, online algorithms, adversarial injections, stochastic injections, stability, memoryless algorithms}
}
Document
APPROX
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Authors: Per Austrin and Aleksa Stanković

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


Abstract
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem.

Cite as

Per Austrin and Aleksa Stanković. Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{austrin_et_al:LIPIcs.APPROX-RANDOM.2019.24,
  author =	{Austrin, Per and Stankovi\'{c}, Aleksa},
  title =	{{Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{24:1--24:17},
  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.2019.24},
  URN =		{urn:nbn:de:0030-drops-112394},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.24},
  annote =	{Keywords: Constraint satisfaction problems, global cardinality constraints, semidefinite programming, inapproximability, Unique Games Conjecture, Max-Cut, Max-2-Sat}
}
Document
RANDOM
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

Authors: Rocco A. Servedio and Li-Yang Tan

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


Abstract
We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F_2 polynomials. Our main results are an epsilon-PRG for the class of size-M depth-d AC^0 circuits with seed length log(M)^{d+O(1)}* log(1/epsilon), and an epsilon-PRG for the class of S-sparse F_2 polynomials with seed length 2^{O(sqrt{log S})}* log(1/epsilon). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently-developed multi-switching lemmas, which are powerful generalizations of Håstad’s switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma - a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family - achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [Impagliazzo et al., 2012] and Håstad [Johan Håstad, 2014]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC^0 and sparse F_2 polynomials.

Cite as

Rocco A. Servedio and Li-Yang Tan. Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 45:1-45:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{servedio_et_al:LIPIcs.APPROX-RANDOM.2019.45,
  author =	{Servedio, Rocco A. and Tan, Li-Yang},
  title =	{{Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{45:1--45:23},
  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.2019.45},
  URN =		{urn:nbn:de:0030-drops-112605},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.45},
  annote =	{Keywords: pseudorandom generators, switching lemmas, circuit complexity, unconditional derandomization}
}
Document
Improved Composition Theorems for Functions and Relations

Authors: Sajin Koroth and Or Meir

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)


Abstract
One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P, i.e., to prove that P is not contained in NC^1. As an approach for this question, Karchmer, Raz and Wigderson [Mauricio Karchmer et al., 1995] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC^{1}. Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [Edmonds et al., 2001], Håstad and Wigderson [Johan Håstad and Avi Wigderson, 1990] and Gavinsky, Meir, Weinstein and Wigderson [Dmitry Gavinsky et al., 2014] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.

Cite as

Sajin Koroth and Or Meir. Improved Composition Theorems for Functions and Relations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{koroth_et_al:LIPIcs.APPROX-RANDOM.2018.48,
  author =	{Koroth, Sajin and Meir, Or},
  title =	{{Improved Composition Theorems for Functions and Relations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.48},
  URN =		{urn:nbn:de:0030-drops-94525},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.48},
  annote =	{Keywords: circuit complexity, communication complexity, KRW conjecture, composition}
}
Document
Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits

Authors: Rocco A. Servedio and Li-Yang Tan

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)


Abstract
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold function) that is fed by S {AND} gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Velickovi{c}, and Wigderson [Michael Luby et al., 1993], who gave the first non-trivial PRG with seed length 2^{O(sqrt{log(S/epsilon)})} that epsilon-fools these circuits. In this work we obtain the first strict improvement of [Michael Luby et al., 1993]'s seed length: we construct a PRG that epsilon-fools size-S {SYM,THR} oAND circuits over {0,1}^n with seed length 2^{O(sqrt{log S})} + polylog(1/epsilon), an exponential (and near-optimal) improvement of the epsilon-dependence of [Michael Luby et al., 1993]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM,THR} o AC^0 circuits. These more general results strengthen previous results of Viola [Viola, 2006] and essentially strengthen more recent results of Lovett and Srinivasan [Lovett and Srinivasan, 2011]. Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan-Wigderson "hardness versus randomness" paradigm [Nisan and Wigderson, 1994]. The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [Johan Håstad, 2014].

Cite as

Rocco A. Servedio and Li-Yang Tan. Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{servedio_et_al:LIPIcs.APPROX-RANDOM.2018.56,
  author =	{Servedio, Rocco A. and Tan, Li-Yang},
  title =	{{Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{56:1--56:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.56},
  URN =		{urn:nbn:de:0030-drops-94601},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.56},
  annote =	{Keywords: Pseudorandom generators, correlation bounds, constant-depth circuits}
}
Document
A Tight Lower Bound for Entropy Flattening

Authors: Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, and Jiapeng Zhang

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)


Abstract
We study entropy flattening: Given a circuit C_X implicitly describing an n-bit source X (namely, X is the output of C_X on a uniform random input), construct another circuit C_Y describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have C_Y evaluate C_X altogether Theta(n^2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit C_Y for entropy flattening that repeatedly queries C_X as an oracle requires Omega(n^2) queries. Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [Johan Håstad et al., 1999; John Rompel, 1990; Thomas Holenstein, 2006; Iftach Haitner et al., 2006; Iftach Haitner et al., 2009; Iftach Haitner et al., 2013; Iftach Haitner et al., 2010; Salil P. Vadhan and Colin Jia Zheng, 2012]. It is also used in reductions between problems complete for statistical zero-knowledge [Tatsuaki Okamoto, 2000; Amit Sahai and Salil P. Vadhan, 1997; Oded Goldreich et al., 1999; Vadhan, 1999]. The Theta(n^2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

Cite as

Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, and Jiapeng Zhang. A Tight Lower Bound for Entropy Flattening. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 23:1-23:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{chen_et_al:LIPIcs.CCC.2018.23,
  author =	{Chen, Yi-Hsiu and G\"{o}\"{o}s, Mika and Vadhan, Salil P. and Zhang, Jiapeng},
  title =	{{A Tight Lower Bound for Entropy Flattening}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{23:1--23:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.23},
  URN =		{urn:nbn:de:0030-drops-88669},
  doi =		{10.4230/LIPIcs.CCC.2018.23},
  annote =	{Keywords: Entropy, One-way function}
}
Document
Bounded Independence vs. Moduli

Authors: Ravi Boppana, Johan Håstad, Chin Ho Lee, and Emanuele Viola

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


Abstract
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.

Cite as

Ravi Boppana, Johan Håstad, Chin Ho Lee, and Emanuele Viola. Bounded Independence vs. Moduli. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 24:1-24:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{boppana_et_al:LIPIcs.APPROX-RANDOM.2016.24,
  author =	{Boppana, Ravi and H\r{a}stad, Johan and Lee, Chin Ho and Viola, Emanuele},
  title =	{{Bounded Independence vs. Moduli}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{24:1--24:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.24},
  URN =		{urn:nbn:de:0030-drops-66475},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.24},
  annote =	{Keywords: Bounded independence, Modulus}
}
Document
Improved NP-Inapproximability for 2-Variable Linear Equations

Authors: Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, and John Wright

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it's possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.

Cite as

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, and John Wright. Improved NP-Inapproximability for 2-Variable Linear Equations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 341-360, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hastad_et_al:LIPIcs.APPROX-RANDOM.2015.341,
  author =	{H\r{a}stad, Johan and Huang, Sangxia and Manokaran, Rajsekar and O’Donnell, Ryan and Wright, John},
  title =	{{Improved NP-Inapproximability for 2-Variable Linear Equations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{341--360},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.341},
  URN =		{urn:nbn:de:0030-drops-53112},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.341},
  annote =	{Keywords: approximability, unique games, linear equation, gadget, linear programming}
}
Document
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 12451)

Authors: Johan Hastad, Andrei Krokhin, and Dániel Marx

Published in: Dagstuhl Reports, Volume 2, Issue 11 (2013)


Abstract
During the past two decades, an impressive array of diverse methods from several different mathematical fields, including algebra, logic, analysis, probability theory, graph theory, and combinatorics, have been used to analyze both the computational complexity and approximabilty of algorithmic tasks related to the constraint satisfaction problem (CSP), as well as the applicability/limitations of algorithmic techniques. The Dagstuhl Seminar 12451 ``The Constraint Satisfaction Problem: Complexity and Approximability'' was aimed at bringing together researchers using all the different techniques in the study of the CSP, so that they can share their insights. This report documents the material presented during the course of the seminar.

Cite as

Johan Hastad, Andrei Krokhin, and Dániel Marx. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 12451). In Dagstuhl Reports, Volume 2, Issue 11, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@Article{hastad_et_al:DagRep.2.11.1,
  author =	{Hastad, Johan and Krokhin, Andrei and Marx, D\'{a}niel},
  title =	{{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 12451)}},
  pages =	{1--19},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{2},
  number =	{11},
  editor =	{Hastad, Johan and Krokhin, Andrei and Marx, D\'{a}niel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.11.1},
  URN =		{urn:nbn:de:0030-drops-39764},
  doi =		{10.4230/DagRep.2.11.1},
  annote =	{Keywords: Constraint satisfaction problem (CSP); Computational complexity; CSP dichotomy conjecture; Hardness of approximation; Unique games conjceture; Fixed-parameter tractability; Descriptive complexity; niversal algebra; Logic; Decomposition methods}
}
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