11 Search Results for "Kumar, Vinayak M."

Document
DOI: 10.4230/LIPIcs.ITCS.2026.73
Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals

Authors: Daniel Grier, Daniel M. Kane, Jackson Morris, Anthony Ostuni, and Kewen Wu

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We construct a family of distributions {𝒟_n}_n with 𝒟_n over {0, 1}ⁿ and a family of depth-7 quantum circuits {C_n}_n such that 𝒟_n is produced exactly by C_n with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance 1 - e^{-Ω(n)} from 𝒟_n. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation. Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.
Cite as

Daniel Grier, Daniel M. Kane, Jackson Morris, Anthony Ostuni, and Kewen Wu. Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{grier_et_al:LIPIcs.ITCS.2026.73,
  author =	{Grier, Daniel and Kane, Daniel M. and Morris, Jackson and Ostuni, Anthony and Wu, Kewen},
  title =	{{Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{73:1--73:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.73},
  URN =		{urn:nbn:de:0030-drops-253607},
  doi =		{10.4230/LIPIcs.ITCS.2026.73},
  annote =	{Keywords: Shallow circuits, sampling, quantum circuits}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.29
The Tape Reconfiguration Problem and Its Consequences for Dominating Set Reconfiguration

Authors: Nicolas Bousquet, Quentin Deschamps, Arnaud Mary, Amer E. Mouawad, and Théo Pierron

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
A dominating set of a graph G = (V,E) is a set of vertices D ⊆ V whose closed neighborhood is V, i.e., N[D] = V. We view a dominating set as a collection of tokens placed on the vertices of D. In the token sliding variant of the Dominating Set Reconfiguration problem (TS-DSR), we seek to transform a source dominating set into a target dominating set in G by sliding tokens along edges, and while maintaining a dominating set all along the transformation. TS-DSR is known to be PSPACE-complete even restricted to graphs of pathwidth w, for some non-explicit constant w and to be XL-complete parameterized by the size k of the solution. The first contribution of this article consists in using a novel approach to provide the first explicit constant for which the TS-DSR problem is PSPACE-complete, a question that was left open in the literature. From a parameterized complexity perspective, the token jumping variant of DSR, i.e., where tokens can jump to arbitrary vertices, is known to be FPT when parameterized by the size of the dominating sets on nowhere dense classes of graphs. But, in contrast, no non-trivial result was known about TS-DSR. We prove that DSR is actually much harder in the sliding model since it is XL-complete when restricted to bounded pathwidth graphs and even when parameterized by k plus the feedback vertex set number of the graph. This gives, for the first time, a difference of behavior between the complexity under token sliding and token jumping for some problem on graphs of bounded treewidth. All our results are obtained using a brand new method, based on the hardness of the so-called Tape Reconfiguration problem, a problem we believe to be of independent interest. We complement these hardness results with a positive result showing that DSR (parameterized by k) in the sliding model is FPT on planar graphs, also answering an open problem from the literature.
Cite as

Nicolas Bousquet, Quentin Deschamps, Arnaud Mary, Amer E. Mouawad, and Théo Pierron. The Tape Reconfiguration Problem and Its Consequences for Dominating Set Reconfiguration. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bousquet_et_al:LIPIcs.ESA.2025.29,
  author =	{Bousquet, Nicolas and Deschamps, Quentin and Mary, Arnaud and Mouawad, Amer E. and Pierron, Th\'{e}o},
  title =	{{The Tape Reconfiguration Problem and Its Consequences for Dominating Set Reconfiguration}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.29},
  URN =		{urn:nbn:de:0030-drops-244974},
  doi =		{10.4230/LIPIcs.ESA.2025.29},
  annote =	{Keywords: combinatorial reconfiguration, parameterized complexity, structural graph parameters, treewidth, dominating set}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.58
Streaming Diameter of High-Dimensional Points

Authors: Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We improve the space bound for streaming approximation of Diameter but also of Farthest Neighbor queries, Minimum Enclosing Ball and its Coreset, in high-dimensional Euclidean spaces. In particular, our deterministic streaming algorithms store 𝒪(ε^{-2}log(1/(ε))) points. This improves by a factor of ε^{-1} the previous space bound of Agarwal and Sharathkumar (SODA 2010), while retaining the state-of-the-art approximation guarantees, such as √2+ε for Diameter or Farthest Neighbor queries, and also offering a simpler and more complete argument. Moreover, we show that storing Ω(ε^{-1}) points is necessary for a streaming (√2+ε)-approximation of Farthest Pair and Farthest Neighbor queries.
Cite as

Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý. Streaming Diameter of High-Dimensional Points. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 58:1-58:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{halldorsson_et_al:LIPIcs.ESA.2025.58,
  author =	{Halld\'{o}rsson, Magn\'{u}s M. and Matsakis, Nicolaos and Vesel\'{y}, Pavel},
  title =	{{Streaming Diameter of High-Dimensional Points}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{58:1--58:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.58},
  URN =		{urn:nbn:de:0030-drops-245263},
  doi =		{10.4230/LIPIcs.ESA.2025.58},
  annote =	{Keywords: streaming algorithm, farthest pair, diameter, minimum enclosing ball, coreset}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.28
Implications of Better PRGs for Permutation Branching Programs

Authors: Dean Doron and William M. Hoza

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study the challenge of derandomizing constant-width standard-order read-once branching programs (ROBPs). Let c ∈ [1, 2) be any constant. We prove that if there are explicit pseudorandom generators (PRGs) for width-6 length-n permutation ROBPs with error 1/n and seed length Õ(log^c n), then there are explicit hitting set generators (HSGs) for width-4 length-n ROBPs with threshold 1/polylog(n) and seed length Õ(log^c n). For context, there are known explicit PRGs that fool constant-width permutation ROBPs with error ε and seed length O(log(n)⋅log(1/ε)) (Koucký, Nimbhorkar, and Pudlák STOC 2011; De CCC 2011; Steinke ECCC 2012). When ε = 1/n, there are known constructions of weighted pseudorandom generators (WPRGs) that fool polynomial-width permutation ROBPs with seed length Õ(log^{3/2} n) (Pyne and Vadhan CCC 2021; Chen, Hoza, Lyu, Tal, and Wu FOCS 2023; Chattopadhyay and Liao ITCS 2024), but unweighted PRGs with seed length o(log² n) remain elusive. Meanwhile, for width-4 ROBPs, there are no known explicit PRGs, WPRGs, or HSGs with seed length o(log²n). Our reduction can be divided into two parts. First, we show that explicit low-error PRGs for width-6 permutation ROBPs with seed length Õ(log^c n) would imply explicit low-error PRGs for width-3 ROBPs with seed length Õ(log^c n). This would improve Meka, Reingold, and Tal’s PRG (STOC 2019), which has seed length o(log²n) only when the error parameter is relatively large. Second, we show that for any w, n, s, and ε, an explicit PRG for width-w ROBPs with error 0.01/n and seed length s would imply an explicit ε-HSG for width-(w + 1) ROBPs with seed length O(s + log(n)⋅log(1/ε)).
Cite as

Dean Doron and William M. Hoza. Implications of Better PRGs for Permutation Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2025.28,
  author =	{Doron, Dean and Hoza, William M.},
  title =	{{Implications of Better PRGs for Permutation Branching Programs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.28},
  URN =		{urn:nbn:de:0030-drops-243946},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.28},
  annote =	{Keywords: hitting set generators, pseudorandom generators, read-once branching programs}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.67
On Sums of INW Pseudorandom Generators

Authors: William M. Hoza and Zelin Lv

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study a new approach for constructing pseudorandom generators (PRGs) that fool constant-width standard-order read-once branching programs (ROBPs). Let X be the n-bit output distribution of the INW PRG (Impagliazzo, Nisan, and Wigderson, STOC 1994), instantiated using expansion parameter λ. We prove that the bitwise XOR of t independent copies of X fools width-w programs with error n^{log(w + 1)} ⋅ (λ⋅log n)^t. Notably, this error bound is meaningful even for relatively large values of λ such as λ = 1/O(log n). Admittedly, our analysis does not yet imply any improvement in the bottom-line overall seed length required for fooling such programs - it just gives a new way of re-proving the well-known O(log² n) bound. Furthermore, we prove that this shortcoming is not an artifact of our analysis, but rather is an intrinsic limitation of our "XOR of INW" approach. That is, no matter how many copies of the INW generator we XOR together, and no matter how we set the expansion parameters, if the generator fools width-3 programs and the proof of correctness does not use any properties of the expander graphs except their spectral expansion, then we prove that the seed length of the generator is inevitably Ω(log² n). Still, we hope that our work might be a step toward constructing near-optimal PRGs fooling constant-width ROBPs. We suggest that one could try running the INW PRG on t correlated seeds, sampled via another PRG, and taking the bitwise XOR of the outputs.
Cite as

William M. Hoza and Zelin Lv. On Sums of INW Pseudorandom Generators. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 67:1-67:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hoza_et_al:LIPIcs.APPROX/RANDOM.2025.67,
  author =	{Hoza, William M. and Lv, Zelin},
  title =	{{On Sums of INW Pseudorandom Generators}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{67:1--67:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.67},
  URN =		{urn:nbn:de:0030-drops-244330},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.67},
  annote =	{Keywords: INW generator, pseudorandomness, space-bounded computation, XOR Lemmas}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.8
Multipass Linear Sketches for Geometric LP-Type Problems

Authors: N. Efe Çekirge, William Gay, and David P. Woodruff

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important applications in machine learning applications such as classification, bioinformatics, and noisy learning. We study LP-type problems in several streaming and distributed big data models, giving ε-approximation linear sketching algorithms with a focus on the high accuracy regime with low dimensionality d, that is, when d < (1/ε)^0.999. Our main result is an O(ds) pass algorithm with O(s(√d/ε)^{3d/s}) ⋅ poly(d, log (1/ε)) space complexity in words, for any parameter s ∈ [1, d log (1/ε)], to solve ε-approximate LP-type problems of O(d) combinatorial and VC dimension. Notably, by taking s = d log (1/ε), we achieve space complexity polynomial in d and polylogarithmic in 1/ε, presenting exponential improvements in 1/ε over current algorithms. We complement our results by showing lower bounds of (1/ε)^Ω(d) for any 1-pass algorithm solving the (1 + ε)-approximation MEB and linear SVM problems, further motivating our multi-pass approach.
Cite as

N. Efe Çekirge, William Gay, and David P. Woodruff. Multipass Linear Sketches for Geometric LP-Type Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cekirge_et_al:LIPIcs.APPROX/RANDOM.2025.8,
  author =	{\c{C}ekirge, N. Efe and Gay, William and Woodruff, David P.},
  title =	{{Multipass Linear Sketches for Geometric LP-Type Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{8:1--8:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.8},
  URN =		{urn:nbn:de:0030-drops-243741},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.8},
  annote =	{Keywords: Streaming, sketching, LP-type problems}
}
Document
DOI: 10.4230/LIPIcs.ITC.2025.1
Amortized Locally Decodable Codes for Insertions and Deletions

Authors: Jeremiah Blocki and Justin Zhang

Published in: LIPIcs, Volume 343, 6th Conference on Information-Theoretic Cryptography (ITC 2025)

Abstract
Locally Decodable Codes (LDCs) are error correcting codes which permit the recovery of any single message symbol with a low number of queries to the codeword (the locality). Traditional LDC tradeoffs between the rate, locality, and error tolerance are undesirable even in relaxed settings where the encoder/decoder share randomness or where the channel is resource-bounded. Recent work by Blocki and Zhang initiated the study of Hamming amortized Locally Decodable Codes (aLDCs), which allow the local decoder to amortize their number of queries over the recovery of a small subset of message symbols. Surprisingly, Blocki and Zhang construct asymptotically ideal (constant rate, constant amortized locality, and constant error tolerance) Hamming aLDCs in private-key and resource-bounded settings. While this result overcame previous barriers and impossibility results for Hamming LDCs, it is not clear whether the techniques extend to Insdel LDCs. Constructing Insdel LDCs which are resilient to insertion and/or deletion errors is known to be even more challenging. For example, Gupta (STOC'24) proved that no Insdel LDC with constant rate and error tolerance exists even in relaxed settings. Our first contribution is to provide a Hamming-to-Insdel compiler which transforms any amortized Hamming LDC that satisfies a particular property (consecutive interval querying) to amortized Insdel LDC while asymptotically preserving the rate, error tolerance and amortized locality. Prior Hamming-to-Insdel compilers of Ostrovsky and Paskin-Cherniavsky (ICITS'15) and Block et al. (FSTTCS'20) worked for arbitrary Hamming LDCs, but incurred an undesirable polylogarithmic blow-up in the locality. Our second contribution is a construction of an ideal amortized Hamming LDC which satisfies our special property (consecutive interval querying) in the relaxed settings where the sender/receiver share randomness or where the channel is resource bounded. Taken together, we obtain ideal Insdel aLDCs in private-key and resource-bounded settings with constant amortized locality, constant rate and constant error tolerance. This result is surprising in light of Gupta’s (STOC'24) impossibility result which demonstrates a strong separation between locality and amortized locality for Insdel LDCs.
Cite as

Jeremiah Blocki and Justin Zhang. Amortized Locally Decodable Codes for Insertions and Deletions. In 6th Conference on Information-Theoretic Cryptography (ITC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 343, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blocki_et_al:LIPIcs.ITC.2025.1,
  author =	{Blocki, Jeremiah and Zhang, Justin},
  title =	{{Amortized Locally Decodable Codes for Insertions and Deletions}},
  booktitle =	{6th Conference on Information-Theoretic Cryptography (ITC 2025)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-385-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{343},
  editor =	{Gilboa, Niv},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2025.1},
  URN =		{urn:nbn:de:0030-drops-243518},
  doi =		{10.4230/LIPIcs.ITC.2025.1},
  annote =	{Keywords: Amortized Locally Decodable Codes, Insertion and Deletion Errors}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.68
New Pseudorandom Generators and Correlation Bounds Using Extractors

Authors: Vinayak M. Kumar

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalization of structured low-degree 𝔽₂-polynomials that we did not have correlation bounds for before. In particular: - We construct a PRG for width-2 poly(n)-length branching programs which read d bits at a time with seed length 2^O(√{log n}) ⋅ d²log²(1/ε). This comes quadratically close to optimal dependence in d and log(1/ε). Improving the dependence on n would imply nontrivial PRGs for log n-degree 𝔽₂-polynomials. The previous PRG by Bogdanov, Dvir, Verbin, and Yehudayoff had an exponentially worse dependence on d with seed length of O(dlog n + d2^dlog(1/ε)). - We provide the first nontrivial (and nearly optimal) correlation bounds and PRGs against size-n^Ω(log n) AC⁰ circuits with either n^{.99} SYM gates (computing an arbitrary symmetric function) or n^{.49} THR gates (computing an arbitrary linear threshold function). This is a generalization of sparse 𝔽₂-polynomials, which can be simulated by an AC⁰ circuit with one parity gate at the top. Previous work of Servedio and Tan only handled n^{.49} SYM gates or n^{.24} THR gates, and previous work of Lovett and Srinivasan only handled polynomial-size circuits. - We give exponentially small correlation bounds against degree-n^O(1) 𝔽₂-polynomials which are set-multilinear over some arbitrary partition of the input into n^{1-O(1)} parts (noting that at n parts, we recover all low degree polynomials). This vastly generalizes correlation bounds against degree-d polynomials which are set-multilinear over a fixed partition into d blocks, which were established by Bhrushundi, Harsha, Hatami, Kopparty, and Kumar. The common technique behind all of these results is to fortify a hard function with the right type of extractor to obtain stronger correlation bounds for more general models of computation. Although this technique has been used in previous work, they rely on the model simplifying drastically under random restrictions. We view our results as a proof of concept that such fortification can be done even for classes that do not enjoy such behavior.
Cite as

Vinayak M. Kumar. New Pseudorandom Generators and Correlation Bounds Using Extractors. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 68:1-68:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kumar:LIPIcs.ITCS.2025.68,
  author =	{Kumar, Vinayak M.},
  title =	{{New Pseudorandom Generators and Correlation Bounds Using Extractors}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{68:1--68:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.68},
  URN =		{urn:nbn:de:0030-drops-226961},
  doi =		{10.4230/LIPIcs.ITCS.2025.68},
  annote =	{Keywords: Pseudorandom Generators, Correlation Bounds, Constant-Depth Circuits}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.18
Tight Correlation Bounds for Circuits Between AC0 and TC0

Authors: Vinayak M. Kumar

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We initiate the study of generalized AC⁰ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight ≥ k (up to negations of the input bits), which we denote GC⁰(k). The gate set of this class includes biased LTFs like the k-OR (outputs 1 iff ≥ k bits are 1) and k-AND (outputs 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC⁰ and TC⁰. We establish a tight multi-switching lemma for GC⁰(k) circuits, which bounds the probability that several depth-2 GC⁰(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC⁰ circuits lifts to depth-d size-s^{.99} GC⁰(.01 log s) circuits with no loss in parameters (other than hidden constants). Our result has the following applications: - Size-2^Ω(n^{1/d}) depth-d GC⁰(Ω(n^{1/d})) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)). - Size-n^Ω(log n) GC⁰(Ω(log² n)) circuits with n^{.249} arbitrary threshold gates or n^{.499} arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). - There is a seed length O((log m)^{d-1}log(m/ε)log log(m)) pseudorandom generator against size-m depth-d GC⁰(log m) circuits, matching the AC⁰ lower bound of Håstad up to a log log m factor (extending a result of Lyu (CCC, 2022)). - Size-m GC⁰(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).
Cite as

Vinayak M. Kumar. Tight Correlation Bounds for Circuits Between AC0 and TC0. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 18:1-18:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kumar:LIPIcs.CCC.2023.18,
  author =	{Kumar, Vinayak M.},
  title =	{{Tight Correlation Bounds for Circuits Between AC0 and TC0}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{18:1--18:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.18},
  URN =		{urn:nbn:de:0030-drops-182885},
  doi =		{10.4230/LIPIcs.CCC.2023.18},
  annote =	{Keywords: AC⁰, TC⁰, Switching Lemma, Lower Bounds, Correlation Bounds, Circuit Complexity}
}
Document
DOI: 10.4230/LITES.8.2.4
A Hybrid Programming Language for Formal Modeling and Verification of Hybrid Systems

Authors: Eduard Kamburjan, Stefan Mitsch, and Reiner Hähnle

Published in: LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2

Abstract
Designing and modeling complex cyber-physical systems (CPS) faces the double challenge of combined discrete-continuous dynamics and concurrent behavior. Existing formal modeling and verification languages for CPS expose the underlying proof search technology. They lack high-level structuring elements and are not efficiently executable. The ensuing modeling gap renders formal CPS models hard to understand and to validate. We propose a high-level programming-based approach to formal modeling and verification of hybrid systems as a hybrid extension of an Active Objects language. Well-structured hybrid active programs and requirements allow automatic, reachability-preserving translation into differential dynamic logic, a logic for hybrid (discrete-continuous) programs. Verification is achieved by discharging the resulting formulas with the theorem prover KeYmaera X. We demonstrate the usability of our approach with case studies.
Cite as

Eduard Kamburjan, Stefan Mitsch, and Reiner Hähnle. A Hybrid Programming Language for Formal Modeling and Verification of Hybrid Systems. In LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2, pp. 04:1-04:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{kamburjan_et_al:LITES.8.2.4,
  author =	{Kamburjan, Eduard and Mitsch, Stefan and H\"{a}hnle, Reiner},
  title =	{{A Hybrid Programming Language for Formal Modeling and Verification of Hybrid Systems}},
  journal =	{Leibniz Transactions on Embedded Systems},
  pages =	{04:1--04:34},
  ISSN =	{2199-2002},
  year =	{2022},
  volume =	{8},
  number =	{2},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES.8.2.4},
  URN =		{urn:nbn:de:0030-drops-192965},
  doi =		{10.4230/LITES.8.2.4},
  annote =	{Keywords: Active Objects, Differential Dynamic Logic, Hybrid Systems}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.48
Pseudobinomiality of the Sticky Random Walk

Authors: Venkatesan Guruswami and Vinayak M. Kumar

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

Abstract
Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number M of marked vertices visited in a long n-step random walk strongly concentrates around the expected n/2 value. Surprisingly, it was recently shown that the parity of M also has exponentially small bias. Is there a common unification of these results? What other statistics about M resemble the binomial distribution (the Hamming weight of a random n-bit string)? To gain insight into such questions, we analyze a simpler model called the sticky random walk. This model is a natural stepping stone towards understanding expander random walks, and we also show that it is a necessary step. The sticky random walk starts with a random bit and then each subsequent bit independently equals the previous bit with probability (1+λ)/2. Here λ is the proxy for the expander’s (second largest) eigenvalue. Using Krawtchouk expansion of functions, we derive several probabilistic results about the sticky random walk. We show an asymptotically tight Θ(λ) bound on the total variation distance between the (Hamming weight of the) sticky walk and the binomial distribution. We prove that the correlation between the majority and parity bit of the sticky walk is bounded by O(n^{-1/4}). This lends hope to unifying Chernoff bounds and parity concentration, as well as establishing other interesting statistical properties, of expander random walks.
Cite as

Venkatesan Guruswami and Vinayak M. Kumar. Pseudobinomiality of the Sticky Random Walk. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{guruswami_et_al:LIPIcs.ITCS.2021.48,
  author =	{Guruswami, Venkatesan and Kumar, Vinayak M.},
  title =	{{Pseudobinomiality of the Sticky Random Walk}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{48:1--48:19},
  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.2021.48},
  URN =		{urn:nbn:de:0030-drops-135870},
  doi =		{10.4230/LIPIcs.ITCS.2021.48},
  annote =	{Keywords: Expander Graphs, Fourier analysis, Markov Chains, Pseudorandomness, Random Walks}
}
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