17 Search Results for "Cohen, Avi"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.31
The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced

Authors: Amnon Ta-Shma and Ron Zadicario

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Numerous works have studied the probability that a length t-1 random walk on an expander is confined to a given rectangle S_1 × … × S_t, providing both upper and lower bounds for this probability. However, when the densities of the sets S_i may depend on the walk length (e.g., when all set are equal and the density is 1-1/t), the currently best known upper and lower bounds are very far from each other. We give an improved confinement lower bound that almost matches the upper bound. We also study the more general question, of how well random walks fool various classes of test functions. Recently, Golowich and Vadhan proved that random walks on λ-expanders fool Boolean, symmetric functions up to a O(λ) error in total variation distance, with no dependence on the labeling bias. Our techniques extend this result to cases not covered by it, e.g., to functions testing confinement to S_1 × … × S_t, where each set S_i either has density ρ or 1-ρ, for arbitrary ρ. Technique-wise, we extend Beck’s framework for analyzing what is often referred to as the "flow" of linear operators, reducing it to bounding the entries of a product of 2×2 matrices.
Cite as

Amnon Ta-Shma and Ron Zadicario. The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tashma_et_al:LIPIcs.APPROX/RANDOM.2024.31,
  author =	{Ta-Shma, Amnon and Zadicario, Ron},
  title =	{{The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{31:1--31:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.31},
  URN =		{urn:nbn:de:0030-drops-210246},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.31},
  annote =	{Keywords: Expander random walks, Expander hitting property}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.41
Hilbert Functions and Low-Degree Randomness Extractors

Authors: Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, and Chao Yan

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
For S ⊆ 𝔽ⁿ, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted h_S(d,𝔽), is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets S of arbitrary finite grids in 𝔽ⁿ with a fixed size |S|. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size |S|. Understanding the smallest values of Hilbert functions is closely related to the study of degree-d closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-d closures of subsets of 𝔽_qⁿ, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-d closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.
Cite as

Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, and Chao Yan. Hilbert Functions and Low-Degree Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 41:1-41:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{golovnev_et_al:LIPIcs.APPROX/RANDOM.2024.41,
  author =	{Golovnev, Alexander and Guo, Zeyu and Hatami, Pooya and Nagargoje, Satyajeet and Yan, Chao},
  title =	{{Hilbert Functions and Low-Degree Randomness Extractors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{41:1--41:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.41},
  URN =		{urn:nbn:de:0030-drops-210345},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.41},
  annote =	{Keywords: Extractors, Dispersers, Circuits, Hilbert Function, Randomness, Low Degree Polynomials}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.61
When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Authors: Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph G together with a linear inner code C₀. Expander codes are Tanner codes whose defining bipartite graph G has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that δ and d₀ must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time? For C₀ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ > 3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ > 2/3-Ω(1) and he also showed that δ > 1/2 is necessary. For general linear code C₀, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d₀ = Ω(cδ^{-2}) is sufficient, where c is the left-degree of G. In this paper, we give a near-optimal solution to the above question for general C₀ by showing that δ d₀ > 3 is sufficient and δ d₀ > 1 is necessary, thereby also significantly improving Dowling-Gao’s result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.
Cite as

Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. When Can an Expander Code Correct Ω(n) Errors in O(n) Time?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.61,
  author =	{Cheng, Kuan and Ouyang, Minghui and Shangguan, Chong and Shen, Yuanting},
  title =	{{When Can an Expander Code Correct \Omega(n) Errors in O(n) Time?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{61:1--61:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.61},
  URN =		{urn:nbn:de:0030-drops-210543},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.61},
  annote =	{Keywords: expander codes, expander graphs, linear-time decoding}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.62
Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree

Authors: Yotam Dikstein and Irit Dinur

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of SL_n(𝔽_q). The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov’s topological overlap constant, and on Dinur and Meshulam’s cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: - We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. - We give a new "spectral" proof for Evra and Kaufman’s local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. - We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.
Cite as

Yotam Dikstein and Irit Dinur. Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dikstein_et_al:LIPIcs.APPROX/RANDOM.2024.62,
  author =	{Dikstein, Yotam and Dinur, Irit},
  title =	{{Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{62:1--62:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.62},
  URN =		{urn:nbn:de:0030-drops-210556},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.62},
  annote =	{Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Coboundary Expansion, Cocycle Expansion, Cosystolic Expansion}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.68
Sparse High Dimensional Expanders via Local Lifts

Authors: Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree high dimensional expanders is known. As a result, our understanding of these objects is limited. The degree of an i-face in an HDX is the number of (i+1)-faces that contain it. In this work we construct complexes whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular k-dimensional HDX X and outputs another regular k-dimensional HDX X̂ with twice as many vertices. While the degree of vertices in X̂ grows, the degree of the (k-1)-faces in X̂ stays the same. As a result, we obtain a new "algebra-free" construction of HDXs whose (k-1)-face degree is bounded. Our construction algorithm is based on a simple and natural generalization of the expander graph construction by Bilu and Linial [Yehonatan Bilu and Nathan Linial, 2006], which build expander graphs using lifts coming from edge signings. Our construction is based on local lifts of high dimensional expanders, where a local lift is a new complex whose top-level links are lifts of the links of the original complex. We demonstrate that a local lift of an HDX is also an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough D, we use this technique to construct families of bounded degree HDXs with links that have diameter ≥ D.
Cite as

Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor. Sparse High Dimensional Expanders via Local Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 68:1-68:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2024.68,
  author =	{Ben Yaacov, Inbar and Dikstein, Yotam and Maor, Gal},
  title =	{{Sparse High Dimensional Expanders via Local Lifts}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{68:1--68:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.68},
  URN =		{urn:nbn:de:0030-drops-210612},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.68},
  annote =	{Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Lifts, Covers, Explicit Constructions, Randomized Constructions, Deterministic Constructions}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.69
Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors

Authors: Kuan Cheng and Ruiyang Wu

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study randomness extractors in AC⁰ and NC¹. For the AC⁰ setting, we give a logspace-uniform construction such that for every k ≥ n/poly log n, ε ≥ 2^{-poly log n}, it can extract from an arbitrary (n, k) source, with a small constant fraction entropy loss, and the seed length is O(log n/(ε)). The seed length and output length are optimal up to constant factors matching the parameters of the best polynomial time construction such as [Guruswami et al., 2009]. The range of k and ε almost meets the lower bound in [Goldreich et al., 2015] and [Cheng and Li, 2018]. We also generalize the main lower bound of [Goldreich et al., 2015] for extractors in AC⁰, showing that when k < n/poly log n, even strong dispersers do not exist in non-uniform AC⁰. For the NC¹ setting, we also give a logspace-uniform extractor construction with seed length O(log n/(ε)) and a small constant fraction entropy loss in the output. It works for every k ≥ O(log² n), ε ≥ 2^{-O(√k)}. Our main techniques include a new error reduction process and a new output stretch process, based on low-depth circuit implementations for mergers, condensers, and somewhere extractors.
Cite as

Kuan Cheng and Ruiyang Wu. Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.69,
  author =	{Cheng, Kuan and Wu, Ruiyang},
  title =	{{Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{69:1--69:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  URN =		{urn:nbn:de:0030-drops-210623},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  annote =	{Keywords: randomness extractor, uniform AC⁰, error reduction, uniform NC¹, disperser}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.CP.2024.37
Computing Small Rainbow Cycle Numbers with SAT Modulo Symmetries (Short Paper)

Authors: Markus Kirchweger and Stefan Szeider

Published in: LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

Abstract
Envy-freeness up to any good (EFX) is a key concept in Computational Social Choice for the fair division of indivisible goods, where no agent envies another’s allocation after removing any single item. A deeper understanding of EFX allocations is facilitated by exploring the rainbow cycle number (R_f(d)), the largest number of independent sets in a certain class of directed graphs. Upper bounds on R_f(d) provide guarantees to the feasibility of EFX allocations (Chaudhury et al., EC 2021). In this work, we precisely compute the numbers R_f(d) for small values of d, employing the SAT modulo Symmetries framework (Kirchweger and Szeider, CP 2021). SAT modulo Symmetries is tailored specifically for the constraint-based isomorph-free generation of combinatorial structures. We provide an efficient encoding for the rainbow cycle number, comparing eager and lazy approaches. To cope with the huge search space, we extend the encoding with invariant pruning, a new method that significantly speeds up computation.
Cite as

Markus Kirchweger and Stefan Szeider. Computing Small Rainbow Cycle Numbers with SAT Modulo Symmetries (Short Paper). In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 37:1-37:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kirchweger_et_al:LIPIcs.CP.2024.37,
  author =	{Kirchweger, Markus and Szeider, Stefan},
  title =	{{Computing Small Rainbow Cycle Numbers with SAT Modulo Symmetries}},
  booktitle =	{30th International Conference on Principles and Practice of Constraint Programming (CP 2024)},
  pages =	{37:1--37:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-336-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{307},
  editor =	{Shaw, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.37},
  URN =		{urn:nbn:de:0030-drops-207221},
  doi =		{10.4230/LIPIcs.CP.2024.37},
  annote =	{Keywords: EFX, rainbow cycle number, SAT modulo Symmetries, combinatorial search}
}
Document
DOI: 10.4230/LIPIcs.ITC.2024.7
Improved Trade-Offs Between Amortization and Download Bandwidth for Linear HSS

Authors: Keller Blackwell and Mary Wootters

Published in: LIPIcs, Volume 304, 5th Conference on Information-Theoretic Cryptography (ITC 2024)

Abstract
A Homomorphic Secret Sharing (HSS) scheme is a secret-sharing scheme that shares a secret x among s servers, and additionally allows an output client to reconstruct some function f(x) using information that can be locally computed by each server. A key parameter in HSS schemes is download rate, which quantifies how much information the output client needs to download from the servers. Often, download rate is improved by amortizing over 𝓁 instances of the problem, making 𝓁 also a key parameter of interest. Recent work [Fosli et al., 2022] established a limit on the download rate of linear HSS schemes for computing low-degree polynomials and constructed schemes that achieve this optimal download rate; their schemes required amortization over 𝓁 = Ω(s log(s)) instances of the problem. Subsequent work [Blackwell and Wootters, 2023] completely characterized linear HSS schemes that achieve optimal download rate in terms of a coding-theoretic notion termed optimal labelweight codes. A consequence of this characterization was that 𝓁 = Ω(s log(s)) is in fact necessary to achieve optimal download rate. In this paper, we characterize all linear HSS schemes, showing that schemes of any download rate are equivalent to a generalization of optimal labelweight codes. This equivalence is constructive and provides a way to obtain an explicit linear HSS scheme from any linear code. Using this characterization, we present explicit linear HSS schemes with slightly sub-optimal rate but with much improved amortization 𝓁 = O(s). Our constructions are based on algebraic geometry codes (specifically Hermitian codes and Goppa codes).
Cite as

Keller Blackwell and Mary Wootters. Improved Trade-Offs Between Amortization and Download Bandwidth for Linear HSS. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blackwell_et_al:LIPIcs.ITC.2024.7,
  author =	{Blackwell, Keller and Wootters, Mary},
  title =	{{Improved Trade-Offs Between Amortization and Download Bandwidth for Linear HSS}},
  booktitle =	{5th Conference on Information-Theoretic Cryptography (ITC 2024)},
  pages =	{7:1--7:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-333-1},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{304},
  editor =	{Aggarwal, Divesh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2024.7},
  URN =		{urn:nbn:de:0030-drops-205156},
  doi =		{10.4230/LIPIcs.ITC.2024.7},
  annote =	{Keywords: Error Correcting Codes, Homomorphic Secret Sharing}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.8
Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries

Authors: Gil Cohen and Tal Yankovitz

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log^{69} n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^{e+o(1)} exist. In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)^{2+o(1)}. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by "vanilla" expander codes which indeed have the necessary property: local testability in the code’s vicinity.
Cite as

Gil Cohen and Tal Yankovitz. Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2024.8,
  author =	{Cohen, Gil and Yankovitz, Tal},
  title =	{{Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.8},
  URN =		{urn:nbn:de:0030-drops-204045},
  doi =		{10.4230/LIPIcs.CCC.2024.8},
  annote =	{Keywords: Relaxed locally decodable codes, Relxaed locally correctable codes, RLCC, RLDC}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.10
Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Authors: Xin Li and Yan Zhong

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).
Cite as

Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.10},
  URN =		{urn:nbn:de:0030-drops-204060},
  doi =		{10.4230/LIPIcs.CCC.2024.10},
  annote =	{Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.32
BPL ⊆ L-AC¹

Authors: Kuan Cheng and Yichuan Wang

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Whether BPL = 𝖫 (which is conjectured to be equal) or even whether BPL ⊆ NL, is a big open problem in theoretical computer science. It is well known that 𝖫 ⊆ NL ⊆ L-AC¹. In this work we show that BPL ⊆ L-AC¹ also holds. Our proof is based on a new iteration method for boosting precision in approximating matrix powering, which is inspired by the Richardson Iteration method developed in a recent line of work [AmirMahdi Ahmadinejad et al., 2020; Edward Pyne and Salil P. Vadhan, 2021; Gil Cohen et al., 2021; William M. Hoza, 2021; Gil Cohen et al., 2023; Aaron (Louie) Putterman and Edward Pyne, 2023; Lijie Chen et al., 2023]. We also improve the algorithm for approximate counting in low-depth L-AC circuits from an additive error setting to a multiplicative error setting.
Cite as

Kuan Cheng and Yichuan Wang. BPL ⊆ L-AC¹. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.CCC.2024.32,
  author =	{Cheng, Kuan and Wang, Yichuan},
  title =	{{BPL ⊆ L-AC¹}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{32:1--32:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.32},
  URN =		{urn:nbn:de:0030-drops-204282},
  doi =		{10.4230/LIPIcs.CCC.2024.32},
  annote =	{Keywords: Randomized Space Complexity, Circuit Complexity, Derandomization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.108
Two-Source and Affine Non-Malleable Extractors for Small Entropy

Authors: Xin Li and Yan Zhong

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Non-malleable extractors are generalizations and strengthening of standard randomness extractors, that are resilient to adversarial tampering. Such extractors have wide applications in cryptography and have become important cornerstones in recent breakthroughs of explicit constructions of two-source extractors and affine extractors for small entropy. However, explicit constructions of non-malleable extractors appear to be much harder than standard extractors. Indeed, in the well-studied models of two-source and affine non-malleable extractors, the previous best constructions only work for entropy rate > 2/3 and 1-γ for some small constant γ > 0 respectively by Li (FOCS' 23). In this paper, we present explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard ones for small entropy. Our main results include: - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k ≥ log^C n and polynomially small error, matching the parameters of standard extractors by Chattopadhyay and Zuckerman (STOC' 16, Annals of Mathematics' 19) and Li (FOCS' 16). - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k = O(log n) and constant error, matching the parameters of standard extractors by Li (FOCS' 23). Our constructions significantly improve previous results, and the parameters (entropy requirement and error) are the best possible without first improving the constructions of standard extractors. In addition, our improved affine non-malleable extractors give strong lower bounds for a certain kind of read-once linear branching programs, recently introduced by Gryaznov, Pudlák, and Talebanfard (CCC' 22) as a generalization of several well studied computational models. These bounds match the previously best-known average-case hardness results given by Chattopadhyay and Liao (CCC' 23) and Li (FOCS' 23), where the branching program size lower bounds are close to optimal, but the explicit functions we use here are different. Our results also suggest a possible deeper connection between non-malleable extractors and standard ones.
Cite as

Xin Li and Yan Zhong. Two-Source and Affine Non-Malleable Extractors for Small Entropy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 108:1-108:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ICALP.2024.108,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Two-Source and Affine Non-Malleable Extractors for Small Entropy}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{108:1--108:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.108},
  URN =		{urn:nbn:de:0030-drops-202512},
  doi =		{10.4230/LIPIcs.ICALP.2024.108},
  annote =	{Keywords: Randomness Extractors, Non-malleable, Two-source, Affine}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.119
Better Sparsifiers for Directed Eulerian Graphs

Authors: Sushant Sachdeva, Anvith Thudi, and Yibin Zhao

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distributions, hitting and commute times, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph G, (1+ε)-sparsifiers with O(nε^{-2}) edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require Ω(nε^{-2}log⁴ n) edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1) We show that for every directed Eulerian graph G→, there exists an Eulerian sparsifier with O(nε^{-2} log² n log²log n + nε^{-4/3}log^{8/3} n) edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2) We give an improved analysis of the constructions based on short-cycle decompositions, giving an m^{1+δ}-time algorithm for any constant δ > 0 for constructing Eulerian sparsifiers with O(nε^{-2}log³ n) edges.
Cite as

Sushant Sachdeva, Anvith Thudi, and Yibin Zhao. Better Sparsifiers for Directed Eulerian Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 119:1-119:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{sachdeva_et_al:LIPIcs.ICALP.2024.119,
  author =	{Sachdeva, Sushant and Thudi, Anvith and Zhao, Yibin},
  title =	{{Better Sparsifiers for Directed Eulerian Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{119:1--119:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.119},
  URN =		{urn:nbn:de:0030-drops-202628},
  doi =		{10.4230/LIPIcs.ICALP.2024.119},
  annote =	{Keywords: Graph algorithms, Linear algebra and computation, Discrepancy theory}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.16
NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors: Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.
Cite as

Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baraskar_et_al:LIPIcs.ICALP.2024.16,
  author =	{Baraskar, Omkar and Dewan, Agrim and Saha, Chandan and Sinha, Pulkit},
  title =	{{NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.16},
  URN =		{urn:nbn:de:0030-drops-201598},
  doi =		{10.4230/LIPIcs.ICALP.2024.16},
  annote =	{Keywords: Equivalence testing, MCSP, sparse polynomials, 3SAT}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.32
Budgeted Dominating Sets in Uncertain Graphs

Authors: Keerti Choudhary, Avi Cohen, N. S. Narayanaswamy, David Peleg, and R. Vijayaragunathan

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
We study the Budgeted Dominating Set (BDS) problem on uncertain graphs, namely, graphs with a probability distribution p associated with the edges, such that an edge e exists in the graph with probability p(e). The input to the problem consists of a vertex-weighted uncertain graph 𝒢 = (V, E, p, ω) and an integer budget (or solution size) k, and the objective is to compute a vertex set S of size k that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of S. We refer to the problem as the Probabilistic Budgeted Dominating Set (PBDS) problem. In this article, we present the following results on the complexity of the PBDS problem. 1) We show that the PBDS problem is NP-complete even when restricted to uncertain trees of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is 𝖶[1]-hard for the budget parameter k, and under the Exponential time hypothesis it cannot be solved in n^o(k) time. 2) We show that if one is willing to settle for (1-ε) approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time. 3) We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is 𝖶[1]-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget k and the treewidth. 4) Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. Our first hardness proof (Thm. 1) makes use of the new problem of k-Subset Σ-Π Maximization (k-SPM), which we believe is of independent interest. We prove its NP-hardness by a reduction from the well-known k-SUM problem, presenting a close relationship between the two problems.
Cite as

Keerti Choudhary, Avi Cohen, N. S. Narayanaswamy, David Peleg, and R. Vijayaragunathan. Budgeted Dominating Sets in Uncertain Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{choudhary_et_al:LIPIcs.MFCS.2021.32,
  author =	{Choudhary, Keerti and Cohen, Avi and Narayanaswamy, N. S. and Peleg, David and Vijayaragunathan, R.},
  title =	{{Budgeted Dominating Sets in Uncertain Graphs}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{32:1--32:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.32},
  URN =		{urn:nbn:de:0030-drops-144723},
  doi =		{10.4230/LIPIcs.MFCS.2021.32},
  annote =	{Keywords: Uncertain graphs, Dominating set, NP-hard, PTAS, treewidth, planar graph}
}
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