18 Search Results for "Kindler, Guy"


Document
APPROX
More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More

Authors: Surendra Ghentiyala and Noah Stephens-Davidowitz

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


Abstract
We expand on recent exciting work of Debris-Alazard, Ducas, and van Woerden [Transactions on Information Theory, 2022], which introduced the notion of basis reduction for codes, in analogy with the extremely successful paradigm of basis reduction for lattices. We generalize DDvW’s LLL algorithm and size-reduction algorithm from codes over 𝔽₂ to codes over 𝔽_q, and we further develop the theory of proper bases. We then show how to instantiate for codes the BKZ and slide-reduction algorithms, which are the two most important generalizations of the LLL algorithm for lattices. Perhaps most importantly, we show a new and very efficient basis-reduction algorithm for codes, called full backward reduction. This algorithm is quite specific to codes and seems to have no analogue in the lattice setting. We prove that this algorithm finds vectors as short as LLL does in the worst case (i.e., within the Griesmer bound) and does so in less time. We also provide both heuristic and empirical evidence that it outperforms LLL in practice, and we give a variant of the algorithm that provably outperforms LLL (in some sense) for random codes. Finally, we explore the promise and limitations of basis reduction for codes. In particular, we show upper and lower bounds on how "good" of a basis a code can have, and we show two additional illustrative algorithms that demonstrate some of the promise and the limitations of basis reduction for codes.

Cite as

Surendra Ghentiyala and Noah Stephens-Davidowitz. More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{ghentiyala_et_al:LIPIcs.APPROX/RANDOM.2024.19,
  author =	{Ghentiyala, Surendra and Stephens-Davidowitz, Noah},
  title =	{{More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{19:1--19:22},
  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.2024.19},
  URN =		{urn:nbn:de:0030-drops-210120},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.19},
  annote =	{Keywords: Linear Codes, Basis Reduction}
}
Document
APPROX
Learning-Augmented Maximum Independent Set

Authors: Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, and Chen Wang

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


Abstract
We study the Maximum Independent Set (MIS) problem on general graphs within the framework of learning-augmented algorithms. The MIS problem is known to be NP-hard and is also NP-hard to approximate to within a factor of n^(1-δ) for any δ > 0. We show that we can break this barrier in the presence of an oracle obtained through predictions from a machine learning model that answers vertex membership queries for a fixed MIS with probability 1/2+ε. In the first setting we consider, the oracle can be queried once per vertex to know if a vertex belongs to a fixed MIS, and the oracle returns the correct answer with probability 1/2 + ε. Under this setting, we show an algorithm that obtains an Õ((√Δ)/ε)-approximation in O(m) time where Δ is the maximum degree of the graph. In the second setting, we allow multiple queries to the oracle for a vertex, each of which is correct with probability 1/2 + ε. For this setting, we show an O(1)-approximation algorithm using O(n/ε²) total queries and Õ(m) runtime.

Cite as

Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, and Chen Wang. Learning-Augmented Maximum Independent Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{braverman_et_al:LIPIcs.APPROX/RANDOM.2024.24,
  author =	{Braverman, Vladimir and Dharangutte, Prathamesh and Shah, Vihan and Wang, Chen},
  title =	{{Learning-Augmented Maximum Independent Set}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{24:1--24:18},
  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.2024.24},
  URN =		{urn:nbn:de:0030-drops-210179},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.24},
  annote =	{Keywords: Learning-augmented algorithms, maximum independent set, graph algorithms}
}
Document
RANDOM
On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs

Authors: Lap Chi Lau and Dante Tjowasi

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


Abstract
Houdré and Tetali defined a class of isoperimetric constants φ_p of graphs for 0 ≤ p ≤ 1, and conjectured a Cheeger-type inequality for φ_(1/2) of the form λ₂ ≲ φ_(1/2) ≲ √λ₂, where λ₂ is the second smallest eigenvalue of the normalized Laplacian matrix. If true, the conjecture would be a strengthening of the hard direction of the classical Cheeger’s inequality. Morris and Peres proved Houdré and Tetali’s conjecture up to an additional log factor, using techniques from evolving sets. We present the following related results on this conjecture. 1) We provide a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed. 2) We match Morris and Peres’s bound using standard spectral arguments. 3) We prove that Houdré and Tetali’s conjecture is true for any constant p strictly bigger than 1/2, which is also a strengthening of the hard direction of Cheeger’s inequality. Furthermore, our results can be extended to directed graphs using Chung’s definition of eigenvalues for directed graphs.

Cite as

Lap Chi Lau and Dante Tjowasi. On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{lau_et_al:LIPIcs.APPROX/RANDOM.2024.36,
  author =	{Lau, Lap Chi and Tjowasi, Dante},
  title =	{{On the Houdr\'{e}-Tetali Conjecture About an Isoperimetric Constant of Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{36:1--36:18},
  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.2024.36},
  URN =		{urn:nbn:de:0030-drops-210295},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.36},
  annote =	{Keywords: Isoperimetric constant, Markov chains, Cheeger’s inequality}
}
Document
RANDOM
Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions

Authors: Hadley Black

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


Abstract
We study monotonicity testing of functions f : {0,1}^d → {0,1} using sample-based algorithms, which are only allowed to observe the value of f on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with exp(Õ(min{(1/ε)√d,d})) samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was Ω(√{exp(d)/ε}) in the small ε parameter regime, when ε = O(d^{-3/2}), due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for ε ≫ d^{-3/2}. We resolve this question, obtaining a nearly tight lower bound of exp(Ω(min{(1/ε)√d,d})) for all ε at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of k-monotonicity testing and learning for functions f : {0,1}^d → [r] is exp(Ω(min{(rk/ε)√d,d})). For testing with one-sided error we show that the sample complexity is exp(Ω(d)). Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of d,k,r,1/ε in the exponent) of exp(Θ̃(min{(rk/ε)√d,d})) on the sample complexity of testing and learning measurable k-monotone functions f : ℝ^d → [r] under product distributions. Our upper bound improves upon the previous bound of exp(Õ(min{(k/ε²)√d,d})) by Harms-Yoshida (ICALP 2022) for Boolean functions (r = 2).

Cite as

Hadley Black. Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{black:LIPIcs.APPROX/RANDOM.2024.37,
  author =	{Black, Hadley},
  title =	{{Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{37:1--37:23},
  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.2024.37},
  URN =		{urn:nbn:de:0030-drops-210308},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.37},
  annote =	{Keywords: Property testing, learning, Boolean functions, monotonicity, k-monotonicity}
}
Document
(Quantum) Complexity of Testing Signed Graph Clusterability

Authors: Kuo-Chin Chen, Simon Apers, and Min-Hsiu Hsieh

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)


Abstract
This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity Õ(N^{1/3}) for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [Adriaens and Apers, 2023]. Second, we prove an Ω̃(√N) classical query lower bound for testing clusterability, which nearly matches the upper bound from [Adriaens and Apers, 2023]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.

Cite as

Kuo-Chin Chen, Simon Apers, and Min-Hsiu Hsieh. (Quantum) Complexity of Testing Signed Graph Clusterability. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{chen_et_al:LIPIcs.TQC.2024.8,
  author =	{Chen, Kuo-Chin and Apers, Simon and Hsieh, Min-Hsiu},
  title =	{{(Quantum) Complexity of Testing Signed Graph Clusterability}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.8},
  URN =		{urn:nbn:de:0030-drops-206786},
  doi =		{10.4230/LIPIcs.TQC.2024.8},
  annote =	{Keywords: Quantum Algorithm, classical Query lower Bound, Graph Property testing}
}
Document
Solving Unique Games over Globally Hypercontractive Graphs

Authors: Mitali Bafna and Dor Minzer

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


Abstract
We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions". Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.

Cite as

Mitali Bafna and Dor Minzer. Solving Unique Games over Globally Hypercontractive Graphs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bafna_et_al:LIPIcs.CCC.2024.3,
  author =	{Bafna, Mitali and Minzer, Dor},
  title =	{{Solving Unique Games over Globally Hypercontractive Graphs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{3:1--3:15},
  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.2024.3},
  URN =		{urn:nbn:de:0030-drops-203996},
  doi =		{10.4230/LIPIcs.CCC.2024.3},
  annote =	{Keywords: unique games, approximation algorithms}
}
Document
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
Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

Authors: Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson

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


Abstract
When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.

Cite as

Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14,
  author =	{B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
  title =	{{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{14:1--14:48},
  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.2024.14},
  URN =		{urn:nbn:de:0030-drops-204100},
  doi =		{10.4230/LIPIcs.CCC.2024.14},
  annote =	{Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem}
}
Document
Information Dissemination via Broadcasts in the Presence of Adversarial Noise

Authors: Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena

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


Abstract
We initiate the study of error correcting codes over the multi-party adversarial broadcast channel. Specifically, we consider the classic information dissemination problem where n parties, each holding an input bit, wish to know each other’s input. For this, they communicate in rounds, where, in each round, one designated party sends a bit to all other parties over a channel governed by an adversary that may corrupt a constant fraction of the received communication. We mention that the dissemination problem was studied in the stochastic noise model since the 80’s. While stochastic noise in multi-party channels has received quite a bit of attention, the case of adversarial noise has largely been avoided, as such channels cannot handle more than a 1/n-fraction of errors. Indeed, this many errors allow an adversary to completely corrupt the incoming or outgoing communication for one of the parties and fail the protocol. Curiously, we show that by eliminating these "trivial" attacks, one can get a simple protocol resilient to a constant fraction of errors. Thus, a model that rules out such attacks is both necessary and sufficient to get a resilient protocol. The main shortcoming of our dissemination protocol is its length: it requires Θ(n²) communication rounds whereas n rounds suffice in the absence of noise. Our main result is a matching lower bound of Ω(n²) on the length of any dissemination protocol in our model. Our proof first "gets rid" of the channel noise by converting it to a form of "input noise", showing that a noisy dissemination protocol implies a (noiseless) protocol for a version of the direct sum gap-majority problem. We conclude the proof with a tight lower bound for the latter problem, which may be of independent interest.

Cite as

Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena. Information Dissemination via Broadcasts in the Presence of Adversarial Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 19:1-19:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{efremenko_et_al:LIPIcs.CCC.2024.19,
  author =	{Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Raz, Ran and Saxena, Raghuvansh R.},
  title =	{{Information Dissemination via Broadcasts in the Presence of Adversarial Noise}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{19:1--19:33},
  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.2024.19},
  URN =		{urn:nbn:de:0030-drops-204159},
  doi =		{10.4230/LIPIcs.CCC.2024.19},
  annote =	{Keywords: Radio Networks, Interactive Coding, Error Correcting Codes}
}
Document
Track A: Algorithms, Complexity and Games
Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH

Authors: Shuangle Li, Bingkai Lin, and Yuwei Liu

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


Abstract
In this paper we present a new gap-creating randomized self-reduction for the parameterized Maximum Likelihood Decoding problem over 𝔽_p (k-MLD_p). The reduction takes a k-MLD_p instance with k⋅ n d-dimensional vectors as input, runs in O(d2^{O(k)}n^{1.01}) time for some computable function f, outputs a (3/2-ε)-Gap-k'-MLD_p instance for any ε > 0, where k' = O(k²log k). Using this reduction, we show that assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can approximate k-MLD_p (and therefore its dual problem k-NCP_p) within factor (3/2-ε) in f(k)⋅ n^{o(√{k/log k})} time for any ε > 0. We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi (ICALP 2018) to amplify the (3/2-ε)-gap to any constant. As a result, we show that assuming ETH, no algorithms can approximate k-NCP_p and k-MDP_p within γ-factor in f(k)⋅ n^{o(k^{ε_γ})} time for some constant ε_γ > 0. Combining with the gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC 2023), we also obtain similar lower bounds for k-MDP_p, k-CVP_p and k-SVP_p. These results improve upon the previous f(k)⋅ n^{Ω(poly log k)} lower bounds for these problems under ETH using reductions by Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).

Cite as

Shuangle Li, Bingkai Lin, and Yuwei Liu. Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 107:1-107:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{li_et_al:LIPIcs.ICALP.2024.107,
  author =	{Li, Shuangle and Lin, Bingkai and Liu, Yuwei},
  title =	{{Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{107:1--107:20},
  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.2024.107},
  URN =		{urn:nbn:de:0030-drops-202500},
  doi =		{10.4230/LIPIcs.ICALP.2024.107},
  annote =	{Keywords: Nearest Codeword Problem, Hardness of Approximations, Fine-grained Complexity, Parameterized Complexity, Minimum Distance Problem, Shortest Vector Problem}
}
Document
Track A: Algorithms, Complexity and Games
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
On the Cut-Query Complexity of Approximating Max-Cut

Authors: Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg

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


Abstract
We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [Rubinstein et al., 2018]. Graph algorithms in this cut query model and other query models have recently been studied for various other problems such as min-cut, connectivity, bipartiteness, and triangle detection. Max-cut in the cut query model can also be viewed as a natural special case of submodular function maximization: on query S ⊆ V, the oracle returns the total weight of the cut between S and V\S. Our first main technical result is a lower bound stating that a deterministic algorithm achieving a c-approximation for any c > 1/2 requires Ω(n) queries. This uses an extension of the cut dimension to rule out approximation (prior work of [Graur et al., 2020] introducing the cut dimension only rules out exact solutions). Secondly, we provide a randomized algorithm with Õ(n) queries that finds a c-approximation for any c < 1. We achieve this using a query-efficient sparsifier for undirected weighted graphs (prior work of [Rubinstein et al., 2018] holds only for unweighted graphs). To complement these results, for most constants c ∈ (0,1], we nail down the query complexity of achieving a c-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at c = 1/2: we design a deterministic algorithm for global c-approximate max-cut in O(log n) queries for any c < 1/2, and show that any randomized algorithm requires Ω(n/log n) queries to find a c-approximate max-cut for any c > 1/2. Additionally, we show that any deterministic algorithm requires Ω(n²) queries to find an exact max-cut (enough to learn the entire graph).

Cite as

Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg. On the Cut-Query Complexity of Approximating Max-Cut. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{plevrakis_et_al:LIPIcs.ICALP.2024.115,
  author =	{Plevrakis, Orestis and Ragavan, Seyoon and Weinberg, S. Matthew},
  title =	{{On the Cut-Query Complexity of Approximating Max-Cut}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{115:1--115:20},
  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.2024.115},
  URN =		{urn:nbn:de:0030-drops-202587},
  doi =		{10.4230/LIPIcs.ICALP.2024.115},
  annote =	{Keywords: query complexity, maximum cut, approximation algorithms, graph sparsification}
}
Document
Improved Monotonicity Testers via Hypercube Embeddings

Authors: Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer

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


Abstract
We show improved monotonicity testers for the Boolean hypercube under the p-biased measure, as well as over the hypergrid [m]ⁿ. Our results are: 1) For any p ∈ (0,1), for the p-biased hypercube we show a non-adaptive tester that makes Õ(√n/ε²) queries, accepts monotone functions with probability 1 and rejects functions that are ε-far from monotone with probability at least 2/3. 2) For all m ∈ ℕ, we show an Õ(√nm³/ε²) query monotonicity tester over [m]ⁿ. We also establish corresponding directed isoperimetric inequalities in these domains, analogous to the isoperimetric inequality in [Subhash Khot et al., 2018]. Previously, the best known tester due to Black, Chakrabarty and Seshadhri [Hadley Black et al., 2018] had Ω(n^{5/6}) query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on m. Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension n into a function over a Boolean cube of a larger dimension n', while preserving its distance from being monotone; an embedding is considered efficient if n' is not much larger than n, and we show how to construct efficient embeddings in the above mentioned settings.

Cite as

Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved Monotonicity Testers via Hypercube Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 25:1-25:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{braverman_et_al:LIPIcs.ITCS.2023.25,
  author =	{Braverman, Mark and Khot, Subhash and Kindler, Guy and Minzer, Dor},
  title =	{{Improved Monotonicity Testers via Hypercube Embeddings}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{25:1--25:24},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.25},
  URN =		{urn:nbn:de:0030-drops-175285},
  doi =		{10.4230/LIPIcs.ITCS.2023.25},
  annote =	{Keywords: Property Testing, Monotonicity Testing, Isoperimetric Inequalities}
}
Document
Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality

Authors: Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra

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


Abstract
We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.

Cite as

Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{kelman_et_al:LIPIcs.ITCS.2021.26,
  author =	{Kelman, Esty and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli},
  title =	{{Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{26:1--26:17},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.26},
  URN =		{urn:nbn:de:0030-drops-135657},
  doi =		{10.4230/LIPIcs.ITCS.2021.26},
  annote =	{Keywords: Fourier Analysis, Hypercontractivity, Log-Sobolev Inequality}
}
Document
Limits of Preprocessing

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

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


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

Cite as

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


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@InProceedings{filmus_et_al:LIPIcs.CCC.2020.17,
  author =	{Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Kindler, Guy},
  title =	{{Limits of Preprocessing}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{17:1--17:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.17},
  URN =		{urn:nbn:de:0030-drops-125697},
  doi =		{10.4230/LIPIcs.CCC.2020.17},
  annote =	{Keywords: circuit, communication complexity, IPPP, preprocessing, PRF, simultaneous messages}
}
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