84 Search Results for "Meka, Raghu"

Volume

LIPIcs, Volume 176

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference

Editors: Jarosław Byrka and Raghu Meka

Document
DOI: 10.4230/LIPIcs.CCC.2024.4
Derandomizing Logspace with a Small Shared Hard Drive

Authors: Edward Pyne

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

Abstract
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
Cite as

Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{pyne:LIPIcs.CCC.2024.4,
  author =	{Pyne, Edward},
  title =	{{Derandomizing Logspace with a Small Shared Hard Drive}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{4:1--4:20},
  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.2024.4},
  URN =		{urn:nbn:de:0030-drops-204006},
  doi =		{10.4230/LIPIcs.CCC.2024.4},
  annote =	{Keywords: Catalytic computation, space-bounded computation, derandomization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.7
Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Authors: Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay

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

Abstract
The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.
Cite as

Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{assadi_et_al:LIPIcs.CCC.2024.7,
  author =	{Assadi, Sepehr and Ghosh, Prantar and Loff, Bruno and Mittal, Parth and Mukhopadhyay, Sagnik},
  title =	{{Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{7:1--7:16},
  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.2024.7},
  URN =		{urn:nbn:de:0030-drops-204035},
  doi =		{10.4230/LIPIcs.CCC.2024.7},
  annote =	{Keywords: Graph streaming, Lower bounds, Communication complexity, k-Cores and degeneracy}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.9
Lifting Dichotomies

Authors: Yaroslav Alekseev, Yuval Filmus, and Alexander Smal

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

Abstract
Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].
Cite as

Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
  title =	{{Lifting Dichotomies}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{9:1--9:18},
  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.2024.9},
  URN =		{urn:nbn:de:0030-drops-204051},
  doi =		{10.4230/LIPIcs.CCC.2024.9},
  annote =	{Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.18
Pseudorandomness, Symmetry, Smoothing: I

Authors: Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola

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

Abstract
We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that - achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. - have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. - rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.
Cite as

Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola. Pseudorandomness, Symmetry, Smoothing: I. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 18:1-18:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{derksen_et_al:LIPIcs.CCC.2024.18,
  author =	{Derksen, Harm and Ivanov, Peter and Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandomness, Symmetry, Smoothing: I}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{18:1--18:27},
  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.2024.18},
  URN =		{urn:nbn:de:0030-drops-204144},
  doi =		{10.4230/LIPIcs.CCC.2024.18},
  annote =	{Keywords: pseudorandomness, k-wise uniform distributions, small-bias distributions, noise, symmetric tests, thresholds, Krawtchouk polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.30
The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise

Authors: Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao

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

Abstract
The class MIP^* of quantum multiprover interactive proof systems with entanglement is much more powerful than its classical counterpart MIP [Babai et al., 1991; Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020]: while MIP = NEXP, the quantum class MIP^* is equal to RE, a class including the halting problem. This is because the provers in MIP^* can share unbounded quantum entanglement. However, recent works [Qin and Yao, 2021; Qin and Yao, 2023] have shown that this advantage is significantly reduced if the provers' shared state contains noise. This paper attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP^*[poly,O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) [Qin and Yao, 2021]. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP^*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continuous.
Cite as

Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dong_et_al:LIPIcs.CCC.2024.30,
  author =	{Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
  title =	{{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{30:1--30:71},
  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.2024.30},
  URN =		{urn:nbn:de:0030-drops-204263},
  doi =		{10.4230/LIPIcs.CCC.2024.30},
  annote =	{Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.12
From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors: Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam

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

Abstract
Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.
Cite as

Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arteche_et_al:LIPIcs.ICALP.2024.12,
  author =	{Arteche, Noel and Khaniki, Erfan and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{From Proof Complexity to Circuit Complexity via Interactive Protocols}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{12:1--12:20},
  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.2024.12},
  URN =		{urn:nbn:de:0030-drops-201557},
  doi =		{10.4230/LIPIcs.ICALP.2024.12},
  annote =	{Keywords: proof complexity, circuit complexity, interactive protocols}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.54
Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs

Authors: Holger Dell, John Lapinskas, and Kitty Meeks

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

Abstract
Consider a query model of computation in which an n-vertex k-hypergraph can be accessed only via its independence oracle or via its colourful independence oracle, and each oracle query may incur a cost depending on the size of the query. Several recent results (Dell and Lapinskas, STOC 2018; Dell, Lapinskas, and Meeks, SODA 2020) give efficient algorithms to approximately count the hypergraph’s edges in the colourful setting. These algorithms immediately imply fine-grained reductions from approximate counting to decision, with overhead only log^Θ(k) n over the running time n^α of the original decision algorithm, for many well-studied problems including k-Orthogonal Vectors, k-SUM, subgraph isomorphism problems including k-Clique and colourful-H, graph motifs, and k-variable first-order model checking. We explore the limits of what is achievable in this setting, obtaining unconditional lower bounds on the oracle cost of algorithms to approximately count the hypergraph’s edges in both the colourful and uncoloured settings. In both settings, we also obtain algorithms which essentially match these lower bounds; in the colourful setting, this requires significant changes to the algorithm of Dell, Lapinskas, and Meeks (SODA 2020) and reduces the total overhead to log^{Θ(k-α)}n. Our lower bound for the uncoloured setting shows that there is no fine-grained reduction from approximate counting to the corresponding uncoloured decision problem (except in the case α ≥ k-1): without an algorithm for the colourful decision problem, we cannot hope to avoid the much larger overhead of roughly n^{(k-α)²/4}. The uncoloured setting has previously been studied for the special case k = 2 (Peled, Ramamoorthy, Rashtchian, Sinha, ITCS 2018; Chen, Levi, and Waingarten, SODA 2020), and our work generalises the existing algorithms and lower bounds for this special case to k > 2 and to oracles with cost.
Cite as

Holger Dell, John Lapinskas, and Kitty Meeks. Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dell_et_al:LIPIcs.ICALP.2024.54,
  author =	{Dell, Holger and Lapinskas, John and Meeks, Kitty},
  title =	{{Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{54:1--54:17},
  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.2024.54},
  URN =		{urn:nbn:de:0030-drops-201977},
  doi =		{10.4230/LIPIcs.ICALP.2024.54},
  annote =	{Keywords: Graph oracles, Fine-grained complexity, Approximate counting, Hypergraphs}
}
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
DOI: 10.4230/LIPIcs.ITCS.2023.60
Loss Minimization Through the Lens Of Outcome Indistinguishability

Authors: Parikshit Gopalan, Lunjia Hu, Michael P. Kim, Omer Reingold, and Udi Wieder

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

Abstract
We present a new perspective on loss minimization and the recent notion of Omniprediction through the lens of Outcome Indistingusihability. For a collection of losses and hypothesis class, omniprediction requires that a predictor provide a loss-minimization guarantee simultaneously for every loss in the collection compared to the best (loss-specific) hypothesis in the class. We present a generic template to learn predictors satisfying a guarantee we call Loss Outcome Indistinguishability. For a set of statistical tests - based on a collection of losses and hypothesis class - a predictor is Loss OI if it is indistinguishable (according to the tests) from Nature’s true probabilities over outcomes. By design, Loss OI implies omniprediction in a direct and intuitive manner. We simplify Loss OI further, decomposing it into a calibration condition plus multiaccuracy for a class of functions derived from the loss and hypothesis classes. By careful analysis of this class, we give efficient constructions of omnipredictors for interesting classes of loss functions, including non-convex losses. This decomposition highlights the utility of a new multi-group fairness notion that we call calibrated multiaccuracy, which lies in between multiaccuracy and multicalibration. We show that calibrated multiaccuracy implies Loss OI for the important set of convex losses arising from Generalized Linear Models, without requiring full multicalibration. For such losses, we show an equivalence between our computational notion of Loss OI and a geometric notion of indistinguishability, formulated as Pythagorean theorems in the associated Bregman divergence. We give an efficient algorithm for calibrated multiaccuracy with computational complexity comparable to that of multiaccuracy. In all, calibrated multiaccuracy offers an interesting tradeoff point between efficiency and generality in the omniprediction landscape.
Cite as

Parikshit Gopalan, Lunjia Hu, Michael P. Kim, Omer Reingold, and Udi Wieder. Loss Minimization Through the Lens Of Outcome Indistinguishability. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gopalan_et_al:LIPIcs.ITCS.2023.60,
  author =	{Gopalan, Parikshit and Hu, Lunjia and Kim, Michael P. and Reingold, Omer and Wieder, Udi},
  title =	{{Loss Minimization Through the Lens Of Outcome Indistinguishability}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{60:1--60:20},
  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.2023.60},
  URN =		{urn:nbn:de:0030-drops-175635},
  doi =		{10.4230/LIPIcs.ITCS.2023.60},
  annote =	{Keywords: Loss Minimization, Indistinguishability}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.21
Random Restrictions and PRGs for PTFs in Gaussian Space

Authors: Zander Kelley and Raghu Meka

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

Abstract
A polynomial threshold function (PTF) f: ℝⁿ → ℝ is a function of the form f(x) = sign(p(x)) where p is a polynomial of degree at most d. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRGs) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a n-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree d PTF. Our main result is a PRG that takes a seed of d^O(1) log(n/ε) log(1/ε)/ε² random bits with output that cannot be distinguished from an n-dimensional gaussian distribution with advantage better than ε by degree d PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seed length of d^O(log d)) in the degree d. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest. Similar results were obtained in [Ryan O'Donnell et al., 2021] (independently and concurrently).
Cite as

Zander Kelley and Raghu Meka. Random Restrictions and PRGs for PTFs in Gaussian Space. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kelley_et_al:LIPIcs.CCC.2022.21,
  author =	{Kelley, Zander and Meka, Raghu},
  title =	{{Random Restrictions and PRGs for PTFs in Gaussian Space}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.21},
  URN =		{urn:nbn:de:0030-drops-165836},
  doi =		{10.4230/LIPIcs.CCC.2022.21},
  annote =	{Keywords: polynomial threshold function, pseudorandom generator, multivariate gaussian}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.27
Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs

Authors: Louis Golowich and Salil Vadhan

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

Abstract
We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. A line of prior work has shown that random walks on expanders with second largest eigenvalue λ fool symmetric functions up to a O(λ) error in total variation distance, but only for the case where the vertices are labeled with symbols from a binary alphabet, and with a suboptimal dependence on the bias of the labeling. We generalize these results to labelings with an arbitrary alphabet, and for the case of binary labelings we achieve an optimal dependence on the labeling bias. We extend our analysis to unify it with and strengthen the expander-walk Chernoff bound. We then show that expander walks fool permutation branching programs up to a O(λ) error in 𝓁₂-distance, and we prove that much stronger bounds hold for programs with a certain structure. We also prove lower bounds to show that our results are tight. To prove our results for symmetric functions, we analyze the Fourier coefficients of the relevant distributions using linear-algebraic techniques. Our analysis for permutation branching programs is likewise linear-algebraic in nature, but also makes use of the recently introduced singular-value approximation notion for matrices (Ahmadinejad et al. 2021).
Cite as

Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{golowich_et_al:LIPIcs.CCC.2022.27,
  author =	{Golowich, Louis and Vadhan, Salil},
  title =	{{Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{27:1--27:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.27},
  URN =		{urn:nbn:de:0030-drops-165893},
  doi =		{10.4230/LIPIcs.CCC.2022.27},
  annote =	{Keywords: Expander graph, Random walk, Pseudorandomness}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.14
Smoothed Analysis of the Komlós Conjecture

Authors: Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the 𝓁_∞ norm of the signed-sum vector is a constant independent of n and d. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors n = ω(d log d). The dependence of n on d is the best possible even in a completely random setting. Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.
Cite as

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Smoothed Analysis of the Komlós Conjecture. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ICALP.2022.14,
  author =	{Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand},
  title =	{{Smoothed Analysis of the Koml\'{o}s Conjecture}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.14},
  URN =		{urn:nbn:de:0030-drops-163556},
  doi =		{10.4230/LIPIcs.ICALP.2022.14},
  annote =	{Keywords: Koml\'{o}s conjecture, smoothed analysis, weighted second moment method, subgaussian coloring}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.13
Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

Authors: Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha

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

Abstract
A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in ℝ^d, find ± signs for each of them such that the signed sum vector along any prefix has a small 𝓁_∞-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an O(√{log d+ log T}) bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk’s bound and consider natural generalizations of prefix discrepancy: - We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in T compared to Banaszczyk’s bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of O(√{log d+ log log T}) with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk’s bound in the worst case. - We also introduce a generalization of the prefix discrepancy problem to arbitrary DAGs. Here, vertices correspond to unit vectors, and the discrepancy constraints correspond to paths on a DAG on T vertices - prefix discrepancy is precisely captured when the DAG is a simple path. We show that an analog of Banaszczyk’s O(√{log d+ log T}) bound continues to hold in this setting for adversarially given unit vectors and that the √{log T} factor is unavoidable for DAGs. We also show that unlike for prefix discrepancy, the dependence on T cannot be improved significantly in the smoothed case for DAGs. - We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. In this problem, the discrepancy constraints are generalized from paths of a DAG to an arbitrary set system. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.
Cite as

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ITCS.2022.13,
  author =	{Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand},
  title =	{{Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{13:1--13:22},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.13},
  URN =		{urn:nbn:de:0030-drops-156092},
  doi =		{10.4230/LIPIcs.ITCS.2022.13},
  annote =	{Keywords: Prefix discrepancy, smoothed analysis, combinatorial vector balancing}
}
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