68 Search Results for "Lovett, Shachar"


Volume

LIPIcs, Volume 234

37th Computational Complexity Conference (CCC 2022)

CCC 2022, July 20-23, 2022, Philadelphia, PA, USA

Editors: Shachar Lovett

Document
Property Testing with Online Adversaries

Authors: Omri Ben-Eliezer, Esty Kelman, Uri Meir, and Sofya Raskhodnikova

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt t data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include t < 1, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.

Cite as

Omri Ben-Eliezer, Esty Kelman, Uri Meir, and Sofya Raskhodnikova. Property Testing with Online Adversaries. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 11:1-11:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2024.11,
  author =	{Ben-Eliezer, Omri and Kelman, Esty and Meir, Uri and Raskhodnikova, Sofya},
  title =	{{Property Testing with Online Adversaries}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{11:1--11:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.11},
  URN =		{urn:nbn:de:0030-drops-195395},
  doi =		{10.4230/LIPIcs.ITCS.2024.11},
  annote =	{Keywords: Linearity testing, low-degree testing, Reed-Muller codes, testing properties of sequences, erasure-resilience, corruption-resilience}
}
Document
On Disperser/Lifting Properties of the Index and Inner-Product Functions

Authors: Paul Beame and Sajin Koroth

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated "lifted" function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [Shachar Lovett et al., 2022] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; - The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N. - The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N. - Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. - Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs.

Cite as

Paul Beame and Sajin Koroth. On Disperser/Lifting Properties of the Index and Inner-Product Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{beame_et_al:LIPIcs.ITCS.2023.14,
  author =	{Beame, Paul and Koroth, Sajin},
  title =	{{On Disperser/Lifting Properties of the Index and Inner-Product Functions}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.14},
  URN =		{urn:nbn:de:0030-drops-175172},
  doi =		{10.4230/LIPIcs.ITCS.2023.14},
  annote =	{Keywords: Decision trees, communication complexity, lifting theorems, proof complexity}
}
Document
Fractional Certificates for Bounded Functions

Authors: Shachar Lovett and Jiapeng Zhang

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold more generally for any bounded function computed by a low degree polynomial. In this work we prove two new results towards establishing this conjecture: first, that any such polynomial has a small fractional certificate complexity; and second, that many inputs have a small sensitive block. We show that these would imply the Aaronson and Ambainis conjecture, assuming a conjectured extension of Talagrand’s concentration inequality. On the technical side, many classical techniques used in the analysis of Boolean functions seem to fail when applied to bounded functions. Here, we develop a new technique, based on a mix of combinatorics, analysis and geometry, and which in part extends a recent technique of Knop et al. (STOC 2021) to bounded functions.

Cite as

Shachar Lovett and Jiapeng Zhang. Fractional Certificates for Bounded Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{lovett_et_al:LIPIcs.ITCS.2023.84,
  author =	{Lovett, Shachar and Zhang, Jiapeng},
  title =	{{Fractional Certificates for Bounded Functions}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{84:1--84:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.84},
  URN =		{urn:nbn:de:0030-drops-175871},
  doi =		{10.4230/LIPIcs.ITCS.2023.84},
  annote =	{Keywords: Aaronson-Ambainis conjecture, fractional block sensitivity, Talagrand inequality}
}
Document
RANDOM
Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Authors: Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang

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


Abstract
Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks. We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the 𝓁₂-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on 𝓁_∞ rather than 𝓁₂-structure.

Cite as

Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang. Eigenstripping, Spectral Decay, and Edge-Expansion on Posets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{gaitonde_et_al:LIPIcs.APPROX/RANDOM.2022.16,
  author =	{Gaitonde, Jason and Hopkins, Max and Kaufman, Tali and Lovett, Shachar and Zhang, Ruizhe},
  title =	{{Eigenstripping, Spectral Decay, and Edge-Expansion on Posets}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{16:1--16:24},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.16},
  URN =		{urn:nbn:de:0030-drops-171381},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.16},
  annote =	{Keywords: High-dimensional expanders, posets, eposets}
}
Document
RANDOM
Lower Bound Methods for Sign-Rank and Their Limitations

Authors: Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, and Rosie Zhao

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


Abstract
The sign-rank of a matrix A with ±1 entries is the smallest rank of a real matrix with the same sign pattern as A. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is at least the VC-dimension; (ii) Forster’s method, which states that sign-rank is at least the inverse of the largest possible average margin among the representations of the matrix by points and half-spaces; (iii) Sign-rank is at least a logarithmic function of the density of the largest monochromatic rectangle. We prove several results regarding the limitations of these methods. - We prove that, qualitatively, the monochromatic rectangle density is the strongest of these three lower bounds. If it fails to provide a super-constant lower bound for the sign-rank of a matrix, then the other two methods will fail as well. - We show that there exist n × n matrices with sign-rank n^Ω(1) for which none of these methods can provide a super-constant lower bound. - We show that sign-rank is at most an exponential function of the deterministic communication complexity with access to an equality oracle. We combine this result with Green and Sanders' quantitative version of Cohen’s idempotent theorem to show that for a large class of sign matrices (e.g., xor-lifts), sign-rank is at most an exponential function of the γ₂ norm of the matrix. We conjecture that this holds for all sign matrices. - Towards answering a question of Linial, Mendelson, Schechtman, and Shraibman regarding the relation between sign-rank and discrepancy, we conjecture that sign-ranks of the ±1 adjacency matrices of hypercube graphs can be arbitrarily large. We prove that none of the three lower bound techniques can resolve this conjecture in the affirmative.

Cite as

Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, and Rosie Zhao. Lower Bound Methods for Sign-Rank and Their Limitations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{hatami_et_al:LIPIcs.APPROX/RANDOM.2022.22,
  author =	{Hatami, Hamed and Hatami, Pooya and Pires, William and Tao, Ran and Zhao, Rosie},
  title =	{{Lower Bound Methods for Sign-Rank and Their Limitations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{22:1--22:24},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.22},
  URN =		{urn:nbn:de:0030-drops-171445},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.22},
  annote =	{Keywords: Average Margin, Communication complexity, margin complexity, monochromatic rectangle, Sign-rank, Unbounded-error communication complexity, VC-dimension}
}
Document
Complete Volume
LIPIcs, Volume 234, CCC 2022, Complete Volume

Authors: Shachar Lovett

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


Abstract
LIPIcs, Volume 234, CCC 2022, Complete Volume

Cite as

37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1-960, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@Proceedings{lovett:LIPIcs.CCC.2022,
  title =	{{LIPIcs, Volume 234, CCC 2022, Complete Volume}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{1--960},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022},
  URN =		{urn:nbn:de:0030-drops-165614},
  doi =		{10.4230/LIPIcs.CCC.2022},
  annote =	{Keywords: LIPIcs, Volume 234, CCC 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Shachar Lovett

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


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lovett:LIPIcs.CCC.2022.0,
  author =	{Lovett, Shachar},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{0:i--0:xvi},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.0},
  URN =		{urn:nbn:de:0030-drops-165621},
  doi =		{10.4230/LIPIcs.CCC.2022.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
The Approximate Degree of Bipartite Perfect Matching

Authors: Gal Beniamini

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


Abstract
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the 𝓁_∞-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching of order n, is Θ̃(n^(3/2)). The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small 𝓁₁-norm - only exponential in Θ(n log n). The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube [Aaronson et al., 2021; Huang, 2019]. We show that the spectral sensitivity of perfect matching is exactly Θ(n^(3/2)).

Cite as

Gal Beniamini. The Approximate Degree of Bipartite Perfect Matching. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{beniamini:LIPIcs.CCC.2022.1,
  author =	{Beniamini, Gal},
  title =	{{The Approximate Degree of Bipartite Perfect Matching}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{1:1--1:26},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.1},
  URN =		{urn:nbn:de:0030-drops-165634},
  doi =		{10.4230/LIPIcs.CCC.2022.1},
  annote =	{Keywords: Bipartite Perfect Matching, Boolean Functions, Approximate Degree}
}
Document
On the Satisfaction Probability of k-CNF Formulas

Authors: Till Tantau

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


Abstract
The satisfaction probability σ(ϕ) := Pr_{β:vars(ϕ) → {0,1}}[β ⊧ ϕ] of a propositional formula ϕ is the likelihood that a random assignment β makes the formula true. We study the complexity of the problem kSAT-PROB_{> δ} = {ϕ is a kCNF formula ∣ σ(ϕ) > δ} for fixed k and δ. While 3SAT-PROB_{> 0} = 3SAT is NP-complete and SAT-PROB}_{> 1/2} is PP-complete, Akmal and Williams recently showed 3SAT-PROB_{> 1/2} ∈ P and 4SAT-PROB_{> 1/2} ∈ NP-complete; but the methods used to prove these striking results stay silent about, say, 4SAT-PROB_{> 3/4}, leaving the computational complexity of kSAT-PROB_{> δ} open for most k and δ. In the present paper we give a complete characterization in the form of a trichotomy: kSAT-PROB_{> δ} lies in AC⁰, is NL-complete, or is NP-complete; and given k and δ we can decide which of the three applies. The proof of the trichotomy hinges on a new order-theoretic insight: Every set of kCNF formulas contains a formula of maximal satisfaction probability. This deceptively simple result allows us to (1) kernelize kSAT-PROB_{≥ δ}, (2) show that the variables of the kernel form a strong backdoor set when the trichotomy states membership in AC⁰ or NL, and (3) prove a locality property by which for every kCNF formula ϕ we have σ(ϕ) ≥ δ iff σ(ψ) ≥ δ for every fixed-size subset ψ of ϕ’s clauses. The locality property will allow us to prove a conjecture of Akmal and Williams: The majority-of-majority satisfaction problem for kCNFS lies in P for all k.

Cite as

Till Tantau. On the Satisfaction Probability of k-CNF Formulas. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{tantau:LIPIcs.CCC.2022.2,
  author =	{Tantau, Till},
  title =	{{On the Satisfaction Probability of k-CNF Formulas}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{2:1--2:27},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.2},
  URN =		{urn:nbn:de:0030-drops-165648},
  doi =		{10.4230/LIPIcs.CCC.2022.2},
  annote =	{Keywords: Satisfaction probability, majority it\{k\}-sat, kernelization, well orderings, locality}
}
Document
Hitting Sets for Regular Branching Programs

Authors: Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne

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


Abstract
We construct improved hitting set generators (HSGs) for ordered (read-once) regular branching programs in two parameter regimes. First, we construct an explicit ε-HSG for unbounded-width regular branching programs with a single accept state with seed length Õ(log n ⋅ log(1/ε)), where n is the length of the program. Second, we construct an explicit ε-HSG for width-w length-n regular branching programs with seed length Õ(log n ⋅ (√{log(1/ε)} + log w) + log(1/ε)). For context, the "baseline" in this area is the pseudorandom generator (PRG) by Nisan (Combinatorica 1992), which fools ordered (possibly non-regular) branching programs with seed length O(log(wn/ε) ⋅ log n). For regular programs, the state-of-the-art PRG, by Braverman, Rao, Raz, and Yehudayoff (FOCS 2010, SICOMP 2014), has seed length Õ(log(w/ε) ⋅ log n), which beats Nisan’s seed length when log(w/ε) = o(log n). Taken together, our two new constructions beat Nisan’s seed length in all parameter regimes except when log w and log(1/ε) are both Ω(log n) (for the construction of HSGs for regular branching programs with a single accept vertex). Extending work by Reingold, Trevisan, and Vadhan (STOC 2006), we furthermore show that an explicit HSG for regular branching programs with a single accept vertex with seed length o(log² n) in the regime log w = Θ(log(1/ε)) = Θ(log n) would imply improved HSGs for general ordered branching programs, which would be a major breakthrough in derandomization. Pyne and Vadhan (CCC 2021) recently obtained such parameters for the special case of permutation branching programs.

Cite as

Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting Sets for Regular Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bogdanov_et_al:LIPIcs.CCC.2022.3,
  author =	{Bogdanov, Andrej and Hoza, William M. and Prakriya, Gautam and Pyne, Edward},
  title =	{{Hitting Sets for Regular Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{3:1--3:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.3},
  URN =		{urn:nbn:de:0030-drops-165658},
  doi =		{10.4230/LIPIcs.CCC.2022.3},
  annote =	{Keywords: Pseudorandomness, hitting set generators, space-bounded computation}
}
Document
Linear Branching Programs and Directional Affine Extractors

Authors: Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard

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


Abstract
A natural model of read-once linear branching programs is a branching program where queries are 𝔽₂ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest. - Proof complexity. We also consider the proof system Res[⊕], which is an extension of resolution with linear queries, and define the regular variant of Res[⊕]. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. If a refutation is regular, we prove that the resulting program is read-once. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a regular Res[⊕] refutation with constant blow up, where the regularity condition comes from the definition of weakly read-once BPs, thus obtaining the equivalence between these proof systems.

Cite as

Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard. Linear Branching Programs and Directional Affine Extractors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{gryaznov_et_al:LIPIcs.CCC.2022.4,
  author =	{Gryaznov, Svyatoslav and Pudl\'{a}k, Pavel and Talebanfard, Navid},
  title =	{{Linear Branching Programs and Directional Affine Extractors}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{4:1--4:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.4},
  URN =		{urn:nbn:de:0030-drops-165664},
  doi =		{10.4230/LIPIcs.CCC.2022.4},
  annote =	{Keywords: Boolean Functions, Average-Case Lower Bounds, AC0\lbrack2\rbrack, Affine Dispersers, Affine Extractors}
}
Document
Quantum Search-To-Decision Reductions and the State Synthesis Problem

Authors: Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen

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


Abstract
It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the de facto computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA problem up to inverse polynomial precision by making one query to a PP decision oracle. We complement this result by showing that QMA-search does not reduce to QMA-decision in polynomial-time, relative to a quantum oracle. We also explore the more general state synthesis problem, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson [Aaronson, 2016], who presented a state synthesis algorithm that makes O(n) queries to a classical oracle to prepare an n-qubit state, and asked if the query complexity could be made sublinear.

Cite as

Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{irani_et_al:LIPIcs.CCC.2022.5,
  author =	{Irani, Sandy and Natarajan, Anand and Nirkhe, Chinmay and Rao, Sujit and Yuen, Henry},
  title =	{{Quantum Search-To-Decision Reductions and the State Synthesis Problem}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{5:1--5:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.5},
  URN =		{urn:nbn:de:0030-drops-165674},
  doi =		{10.4230/LIPIcs.CCC.2022.5},
  annote =	{Keywords: Search-to-decision, state synthesis, quantum computing}
}
Document
Almost Polynomial Factor Inapproximability for Parameterized k-Clique

Authors: Karthik C. S. and Subhash Khot

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


Abstract
The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k) ⋅ poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem. Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]≠FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k) = O(1)) FPT-approximation algorithms under W[1]≠FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every F(k) = k^{1/H(k)} factor FPT-approximation algorithm for any increasing computable function H (for example H(k) = log^∗ k). Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.

Cite as

Karthik C. S. and Subhash Khot. Almost Polynomial Factor Inapproximability for Parameterized k-Clique. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{karthikc.s._et_al:LIPIcs.CCC.2022.6,
  author =	{Karthik C. S. and Khot, Subhash},
  title =	{{Almost Polynomial Factor Inapproximability for Parameterized k-Clique}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{6:1--6:21},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.6},
  URN =		{urn:nbn:de:0030-drops-165680},
  doi =		{10.4230/LIPIcs.CCC.2022.6},
  annote =	{Keywords: Parameterized Complexity, k-clique, Hardness of Approximation}
}
Document
𝓁_p-Spread and Restricted Isometry Properties of Sparse Random Matrices

Authors: Venkatesan Guruswami, Peter Manohar, and Jonathan Mosheiff

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


Abstract
Random subspaces X of ℝⁿ of dimension proportional to n are, with high probability, well-spread with respect to the 𝓁₂-norm. Namely, every nonzero x ∈ X is "robustly non-sparse" in the following sense: x is ε ‖x‖₂-far in 𝓁₂-distance from all δ n-sparse vectors, for positive constants ε, δ bounded away from 0. This "𝓁₂-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to X being a Euclidean section of the 𝓁₁ unit ball. Explicit 𝓁₂-spread subspaces of dimension Ω(n), however, are unknown, and the best known explicit constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of certain sparse matrices. Motivated by this, we study the spread properties of the kernels of sparse random matrices. We prove that with high probability such subspaces contain vectors x that are o(1)⋅‖x‖₂-close to o(n)-sparse with respect to the 𝓁₂-norm, and in particular are not 𝓁₂-spread. This is strikingly different from the case of random LDPC codes, whose distance is asymptotically almost as good as that of (dense) random linear codes. On the other hand, for p < 2 we prove that such subspaces are 𝓁_p-spread with high probability. The spread property of sparse random matrices thus exhibits a threshold behavior at p = 2. Our proof for p < 2 moreover shows that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the 𝓁_p norm, and in fact this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the 𝓁₁ norm [Berinde et al., 2008]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of 𝓁_p-RIP matrices for 1 ≤ p < p₀, where 1 < p₀ < 2 is an absolute constant.

Cite as

Venkatesan Guruswami, Peter Manohar, and Jonathan Mosheiff. 𝓁_p-Spread and Restricted Isometry Properties of Sparse Random Matrices. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{guruswami_et_al:LIPIcs.CCC.2022.7,
  author =	{Guruswami, Venkatesan and Manohar, Peter and Mosheiff, Jonathan},
  title =	{{𝓁\underlinep-Spread and Restricted Isometry Properties of Sparse Random Matrices}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{7:1--7:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.7},
  URN =		{urn:nbn:de:0030-drops-165695},
  doi =		{10.4230/LIPIcs.CCC.2022.7},
  annote =	{Keywords: Spread Subspaces, Euclidean Sections, Restricted Isometry Property, Sparse Matrices}
}
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