99 Search Results for "Lee, James R."

Volume

LIPIcs, Volume 185

12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

ITCS 2021, January 6-8, 2021, Virtual Conference

Editors: James R. Lee

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.10
Hyperbolic Concentration, Anti-Concentration, and Discrepancy

Authors: Zhao Song and Ruizhe Zhang

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

Abstract
Chernoff bound is a fundamental tool in theoretical computer science. It has been extensively used in randomized algorithm design and stochastic type analysis. Discrepancy theory, which deals with finding a bi-coloring of a set system such that the coloring of each set is balanced, has a huge number of applications in approximation algorithms design. Chernoff bound [Che52] implies that a random bi-coloring of any set system with n sets and n elements will have discrepancy O(√{n log n}) with high probability, while the famous result by Spencer [Spe85] shows that there exists an O(√n) discrepancy solution. The study of hyperbolic polynomials dates back to the early 20th century when used to solve PDEs by Gårding [Går59]. In recent years, more applications are found in control theory, optimization, real algebraic geometry, and so on. In particular, the breakthrough result by Marcus, Spielman, and Srivastava [MSS15] uses the theory of hyperbolic polynomials to prove the Kadison-Singer conjecture [KS59], which is closely related to discrepancy theory. In this paper, we present a list of new results for hyperbolic polynomials: - We show two nearly optimal hyperbolic Chernoff bounds: one for Rademacher sum of arbitrary vectors and another for random vectors in the hyperbolic cone. - We show a hyperbolic anti-concentration bound. - We generalize the hyperbolic Kadison-Singer theorem [Brä18] for vectors in sub-isotropic position, and prove a hyperbolic Spencer theorem for any constant hyperbolic rank vectors. The classical matrix Chernoff and discrepancy results are based on determinant polynomial which is a special case of hyperbolic polynomials. To the best of our knowledge, this paper is the first work that shows either concentration or anti-concentration results for hyperbolic polynomials. We hope our findings provide more insights into hyperbolic and discrepancy theories.
Cite as

Zhao Song and Ruizhe Zhang. Hyperbolic Concentration, Anti-Concentration, and Discrepancy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{song_et_al:LIPIcs.APPROX/RANDOM.2022.10,
  author =	{Song, Zhao and Zhang, Ruizhe},
  title =	{{Hyperbolic Concentration, Anti-Concentration, and Discrepancy}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{10:1--10:19},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-171324},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.10},
  annote =	{Keywords: Hyperbolic polynomial, Chernoff bound, Concentration, Discrepancy theory, Anti-concentration}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.60
Multiscale Entropic Regularization for MTS on General Metric Spaces

Authors: Farzam Ebrahimnejad and James R. Lee

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

Abstract
We present an O((log n)²)-competitive algorithm for metrical task systems (MTS) on any n-point metric space that is also 1-competitive for service costs. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, and Lee (2019) and the refined competitive ratios obtained by Coester and Lee (2019). Those algorithms work by first randomly embedding the metric space into an ultrametric and then solving MTS there. In contrast, our algorithm is cast as regularized gradient descent where the regularizer is a multiscale metric entropy defined directly on the metric space. This answers an open question of Bubeck (Highlights of Algorithms, 2019).
Cite as

Farzam Ebrahimnejad and James R. Lee. Multiscale Entropic Regularization for MTS on General Metric Spaces. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ebrahimnejad_et_al:LIPIcs.ITCS.2022.60,
  author =	{Ebrahimnejad, Farzam and Lee, James R.},
  title =	{{Multiscale Entropic Regularization for MTS on General Metric Spaces}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{60:1--60:21},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.60},
  URN =		{urn:nbn:de:0030-drops-156568},
  doi =		{10.4230/LIPIcs.ITCS.2022.60},
  annote =	{Keywords: Metrical task systems, online algorithms, metric embeddings, convex optimization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.90
SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization

Authors: Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√{k(n-k)}) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function. Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf-Yuen conjecture and prove that the SoS rank for SQFs is at most O(√{nk}log(n)). We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√nlog(n)) SoS certificate for this problem.
Cite as

Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth. SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 90:1-90:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kurpisz_et_al:LIPIcs.ICALP.2021.90,
  author =	{Kurpisz, Adam and Potechin, Aaron and Wirth, Elias Samuel},
  title =	{{SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{90:1--90:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.90},
  URN =		{urn:nbn:de:0030-drops-141595},
  doi =		{10.4230/LIPIcs.ICALP.2021.90},
  annote =	{Keywords: symmetric quadratic functions, SoS certificate, hypercube optimization, semidefinite programming}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.111
Fourier Conjectures, Correlation Bounds, and Majority

Authors: Emanuele Viola

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don't, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.
Cite as

Emanuele Viola. Fourier Conjectures, Correlation Bounds, and Majority. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 111:1-111:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{viola:LIPIcs.ICALP.2021.111,
  author =	{Viola, Emanuele},
  title =	{{Fourier Conjectures, Correlation Bounds, and Majority}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{111:1--111:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.111},
  URN =		{urn:nbn:de:0030-drops-141806},
  doi =		{10.4230/LIPIcs.ICALP.2021.111},
  annote =	{Keywords: Fourier analysis, polynomials, Majority, correlation, lower bound, conjectures}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.ITCS.2021
LIPIcs, Volume 185, ITCS 2021, Complete Volume

Authors: James R. Lee

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
LIPIcs, Volume 185, ITCS 2021, Complete Volume
Cite as

12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 1-1550, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{lee:LIPIcs.ITCS.2021,
  title =	{{LIPIcs, Volume 185, ITCS 2021, Complete Volume}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{1--1550},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021},
  URN =		{urn:nbn:de:0030-drops-135381},
  doi =		{10.4230/LIPIcs.ITCS.2021},
  annote =	{Keywords: LIPIcs, Volume 185, ITCS 2021, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.ITCS.2021.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: James R. Lee

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

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

12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lee:LIPIcs.ITCS.2021.0,
  author =	{Lee, James R.},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{0:i--0:xiv},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.0},
  URN =		{urn:nbn:de:0030-drops-135397},
  doi =		{10.4230/LIPIcs.ITCS.2021.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.1
The Entropy of Lies: Playing Twenty Questions with a Liar

Authors: Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
"Twenty questions" is a guessing game played by two players: Bob thinks of an integer between 1 and n, and Alice’s goal is to recover it using a minimal number of Yes/No questions. Shannon’s entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let μ be a distribution over [n], then the average number of questions used by an optimal strategy that recovers x∼ μ is between H(μ) and H(μ)+1. We consider an extension of this game where at most k questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly H(μ) + k H_2(μ) questions, where H_2(μ) = ∑_x μ(x)log log 1/μ(x). This also generalizes a result by Rivest et al. (1980) for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form "x ≤ c?" for c ∈ [n]. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.
Cite as

Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran. The Entropy of Lies: Playing Twenty Questions with a Liar. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dagan_et_al:LIPIcs.ITCS.2021.1,
  author =	{Dagan, Yuval and Filmus, Yuval and Kane, Daniel and Moran, Shay},
  title =	{{The Entropy of Lies: Playing Twenty Questions with a Liar}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.1},
  URN =		{urn:nbn:de:0030-drops-135400},
  doi =		{10.4230/LIPIcs.ITCS.2021.1},
  annote =	{Keywords: entropy, twenty questions, algorithms, sorting}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.2
Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)

Authors: Russell Impagliazzo and Sam McGuire

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Computational pseudorandomness studies the extent to which a random variable Z looks like the uniform distribution according to a class of tests ℱ. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a high entropy distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class ℱ is closed under taking majorities. This equivalence constitutes (essentially) the so-called dense model theorem of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao’s proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where ℱ is not closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is false.
Cite as

Russell Impagliazzo and Sam McGuire. Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{impagliazzo_et_al:LIPIcs.ITCS.2021.2,
  author =	{Impagliazzo, Russell and McGuire, Sam},
  title =	{{Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{2:1--2:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.2},
  URN =		{urn:nbn:de:0030-drops-135417},
  doi =		{10.4230/LIPIcs.ITCS.2021.2},
  annote =	{Keywords: Computational entropy, dense model theorem, coin problem}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.3
Algorithmic Persuasion with Evidence

Authors: Martin Hoefer, Pasin Manurangsi, and Alexandros Psomas

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether or not the receiver takes the action. The receiver’s utility depends on both the action as well as the sender’s private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.
Cite as

Martin Hoefer, Pasin Manurangsi, and Alexandros Psomas. Algorithmic Persuasion with Evidence. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hoefer_et_al:LIPIcs.ITCS.2021.3,
  author =	{Hoefer, Martin and Manurangsi, Pasin and Psomas, Alexandros},
  title =	{{Algorithmic Persuasion with Evidence}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{3:1--3:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.3},
  URN =		{urn:nbn:de:0030-drops-135420},
  doi =		{10.4230/LIPIcs.ITCS.2021.3},
  annote =	{Keywords: Bayesian Persuasion, Semidefinite Programming, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.4
The Complexity of Finding Fair Independent Sets in Cycles

Authors: Ishay Haviv

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Let G be a cycle graph and let V₁,…,V_m be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if |S ∩ V_i| ≥ 1/2⋅|V_i| - 1 for all i ∈ [m]. It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al., A Journey through Discrete Math., 2017). We prove that the problem of finding such an independent set is PPA-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is PPA-complete as well. The work is motivated by the computational aspects of the "cycle plus triangles" problem and of its extensions.
Cite as

Ishay Haviv. The Complexity of Finding Fair Independent Sets in Cycles. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haviv:LIPIcs.ITCS.2021.4,
  author =	{Haviv, Ishay},
  title =	{{The Complexity of Finding Fair Independent Sets in Cycles}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.4},
  URN =		{urn:nbn:de:0030-drops-135431},
  doi =		{10.4230/LIPIcs.ITCS.2021.4},
  annote =	{Keywords: Fair independent sets in cycles, the complexity class \{PPA\}, Schrijver graphs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.5
Sharp Threshold Rates for Random Codes

Authors: Venkatesan Guruswami, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Suppose that 𝒫 is a property that may be satisfied by a random code C ⊂ Σⁿ. For example, for some p ∈ (0,1), 𝒫 might be the property that there exist three elements of C that lie in some Hamming ball of radius pn. We say that R^* is the threshold rate for 𝒫 if a random code of rate R^* + ε is very likely to satisfy 𝒫, while a random code of rate R^* - ε is very unlikely to satisfy 𝒫. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably "symmetric." For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property 𝒫 above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general.
Cite as

Venkatesan Guruswami, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters. Sharp Threshold Rates for Random Codes. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{guruswami_et_al:LIPIcs.ITCS.2021.5,
  author =	{Guruswami, Venkatesan and Mosheiff, Jonathan and Resch, Nicolas and Silas, Shashwat and Wootters, Mary},
  title =	{{Sharp Threshold Rates for Random Codes}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.5},
  URN =		{urn:nbn:de:0030-drops-135446},
  doi =		{10.4230/LIPIcs.ITCS.2021.5},
  annote =	{Keywords: Coding theory, Random codes, Sharp thresholds}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.6
Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation

Authors: Cameron Musco, Christopher Musco, and David P. Woodruff

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
In the masked low-rank approximation problem, one is given data matrix A ∈ ℝ^{n × n} and binary mask matrix W ∈ {0,1}^{n × n}. The goal is to find a rank-k matrix L for which: cost(L) := ∑_{i=1}^n ∑_{j=1}^n W_{i,j} ⋅ (A_{i,j} - L_{i,j})² ≤ OPT + ε ‖A‖_F², where OPT = min_{rank-k L̂} cost(L̂) and ε is a given error parameter. Depending on the choice of W, the above problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, low-rank recovery from monotone missing data, and a number of other important problems. Many of these problems are NP-hard, and while algorithms with provable guarantees are known in some cases, they either 1) run in time n^Ω(k²/ε) or 2) make strong assumptions, for example, that A is incoherent or that the entries in W are chosen independently and uniformly at random. In this work, we show that a common polynomial time heuristic, which simply sets A to 0 where W is 0, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank k' > k depending on the public coin partition number of W, the heuristic outputs rank-k' L with cost(L) ≤ OPT + ε ‖A‖_F². This partition number is in turn bounded by the randomized communication complexity of W, when interpreted as a two-player communication matrix. For many important cases, including all those listed above, this yields bicriteria approximation guarantees with rank k' = k ⋅ poly(log n/ε). Beyond this result, we show that different notions of communication complexity yield bicriteria algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.
Cite as

Cameron Musco, Christopher Musco, and David P. Woodruff. Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{musco_et_al:LIPIcs.ITCS.2021.6,
  author =	{Musco, Cameron and Musco, Christopher and Woodruff, David P.},
  title =	{{Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.6},
  URN =		{urn:nbn:de:0030-drops-135452},
  doi =		{10.4230/LIPIcs.ITCS.2021.6},
  annote =	{Keywords: low-rank approximation, communication complexity, weighted low-rank approximation, bicriteria approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.7
Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Authors: William M. Hoza, Edward Pyne, and Salil Vadhan

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We prove that the Impagliazzo-Nisan-Wigderson [Impagliazzo et al., 1994] pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of Õ(log d + log n ⋅ log(1/ε)), assuming the program has only one accepting vertex in the final layer. Here, n is the length of the program, d is the degree (equivalently, the alphabet size), and ε is the error of the PRG. In contrast, we show that a randomly chosen generator requires seed length Ω(n log d) to fool such unbounded-width programs. Thus, this is an unusual case where an explicit construction is "better than random." Except when the program’s width w is very small, this is an improvement over prior work. For example, when w = poly(n) and d = 2, the best prior PRG for permutation branching programs was simply Nisan’s PRG [Nisan, 1992], which fools general ordered branching programs with seed length O(log(wn/ε) log n). We prove a seed length lower bound of Ω̃(log d + log n ⋅ log(1/ε)) for fooling these unbounded-width programs, showing that our seed length is near-optimal. In fact, when ε ≤ 1/log n, our seed length is within a constant factor of optimal. Our analysis of the INW generator uses the connection between the PRG and the derandomized square of Rozenman and Vadhan [Rozenman and Vadhan, 2005] and the recent analysis of the latter in terms of unit-circle approximation by Ahmadinejad et al. [Ahmadinejad et al., 2020].
Cite as

William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom Generators for Unbounded-Width Permutation Branching Programs. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hoza_et_al:LIPIcs.ITCS.2021.7,
  author =	{Hoza, William M. and Pyne, Edward and Vadhan, Salil},
  title =	{{Pseudorandom Generators for Unbounded-Width Permutation Branching Programs}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.7},
  URN =		{urn:nbn:de:0030-drops-135464},
  doi =		{10.4230/LIPIcs.ITCS.2021.7},
  annote =	{Keywords: Pseudorandom generators, permutation branching programs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.8
Pipeline Interventions

Authors: Eshwar Ram Arunachaleswaran, Sampath Kannan, Aaron Roth, and Juba Ziani

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We introduce the pipeline intervention problem, defined by a layered directed acyclic graph and a set of stochastic matrices governing transitions between successive layers. The graph is a stylized model for how people from different populations are presented opportunities, eventually leading to some reward. In our model, individuals are born into an initial position (i.e. some node in the first layer of the graph) according to a fixed probability distribution, and then stochastically progress through the graph according to the transition matrices, until they reach a node in the final layer of the graph; each node in the final layer has a reward associated with it. The pipeline intervention problem asks how to best make costly changes to the transition matrices governing people’s stochastic transitions through the graph, subject to a budget constraint. We consider two objectives: social welfare maximization, and a fairness-motivated maximin objective that seeks to maximize the value to the population (starting node) with the least expected value. We consider two variants of the maximin objective that turn out to be distinct, depending on whether we demand a deterministic solution or allow randomization. For each objective, we give an efficient approximation algorithm (an additive FPTAS) for constant width networks. We also tightly characterize the "price of fairness" in our setting: the ratio between the highest achievable social welfare and the social welfare consistent with a maximin optimal solution. Finally we show that for polynomial width networks, even approximating the maximin objective to any constant factor is NP hard, even for networks with constant depth. This shows that the restriction on the width in our positive results is essential.
Cite as

Eshwar Ram Arunachaleswaran, Sampath Kannan, Aaron Roth, and Juba Ziani. Pipeline Interventions. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arunachaleswaran_et_al:LIPIcs.ITCS.2021.8,
  author =	{Arunachaleswaran, Eshwar Ram and Kannan, Sampath and Roth, Aaron and Ziani, Juba},
  title =	{{Pipeline Interventions}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.8},
  URN =		{urn:nbn:de:0030-drops-135478},
  doi =		{10.4230/LIPIcs.ITCS.2021.8},
  annote =	{Keywords: Interventions for fairness, fairness in navigating life paths, social welfare, maximin welfare, budget-constrained optimization, hardness of approximation}
}
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