6 Search Results for "Bhattiprolu, Vijay"


Document
Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem

Authors: Vijay Bhattiprolu, Euiwoong Lee, and Madhur Tulsiani

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


Abstract
Grothendieck’s inequality [Grothendieck, 1953] states that there is an absolute constant K > 1 such that for any n× n matrix A, ‖A‖_{∞→1} := max_{s,t ∈ {± 1}ⁿ}∑_{i,j} A[i,j]⋅s(i)⋅t(j) ≥ 1/K ⋅ max_{u_i,v_j ∈ S^{n-1}}∑_{i,j} A[i,j]⋅⟨u_i,v_j⟩. In addition to having a tremendous impact on Banach space theory, this inequality has found applications in several unrelated fields like quantum information, regularity partitioning, communication complexity, etc. Let K_G (known as Grothendieck’s constant) denote the smallest constant K above. Grothendieck’s inequality implies that a natural semidefinite programming relaxation obtains a constant factor approximation to ‖A‖_{∞ → 1}. The exact value of K_G is yet unknown with the best lower bound (1.67…) being due to Reeds and the best upper bound (1.78…) being due to Braverman, Makarychev, Makarychev and Naor [Braverman et al., 2013]. In contrast, the little Grothendieck inequality states that under the assumption that A is PSD the constant K above can be improved to π/2 and moreover this is tight. The inapproximability of ‖A‖_{∞ → 1} has been studied in several papers culminating in a tight UGC-based hardness result due to Raghavendra and Steurer (remarkably they achieve this without knowing the value of K_G). Briet, Regev and Saket [Briët et al., 2015] proved tight NP-hardness of approximating the little Grothendieck problem within π/2, based on a framework by Guruswami, Raghavendra, Saket and Wu [Guruswami et al., 2016] for bypassing UGC for geometric problems. This also remained the best known NP-hardness for the general Grothendieck problem due to the nature of the Guruswami et al. framework, which utilized a projection operator onto the degree-1 Fourier coefficients of long code encodings, which naturally yielded a PSD matrix A. We show how to extend the above framework to go beyond the degree-1 Fourier coefficients, using the global structure of optimal solutions to the Grothendieck problem. As a result, we obtain a separation between the NP-hardness results for the two problems, obtaining an inapproximability result for the Grothendieck problem, of a factor π/2 + ε₀ for a fixed constant ε₀ > 0.

Cite as

Vijay Bhattiprolu, Euiwoong Lee, and Madhur Tulsiani. Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bhattiprolu_et_al:LIPIcs.ITCS.2022.22,
  author =	{Bhattiprolu, Vijay and Lee, Euiwoong and Tulsiani, Madhur},
  title =	{{Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{22:1--22:17},
  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.2022.22},
  URN =		{urn:nbn:de:0030-drops-156186},
  doi =		{10.4230/LIPIcs.ITCS.2022.22},
  annote =	{Keywords: Grothendieck’s Inequality, Hardness of Approximation, Semidefinite Programming, Optimization}
}
Document
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-dev.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}
}
Document
RANDOM
A Simpler Strong Refutation of Random k-XOR

Authors: Kwangjun Ahn

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
Strong refutation of random CSPs is a fundamental question in theoretical computer science that has received particular attention due to the long-standing gap between the information-theoretic limit and the computational limit. This gap is recently bridged by Raghavendra, Rao and Schramm where they study sub-exponential algorithms for the regime between the two limits. In this work, we take a simpler approach to their algorithms and analyses.

Cite as

Kwangjun Ahn. A Simpler Strong Refutation of Random k-XOR. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{ahn:LIPIcs.APPROX/RANDOM.2020.2,
  author =	{Ahn, Kwangjun},
  title =	{{A Simpler Strong Refutation of Random k-XOR}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.2},
  URN =		{urn:nbn:de:0030-drops-126053},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.2},
  annote =	{Keywords: Strong refutation, Random k-XOR, Spectral method, Trace power method}
}
Document
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

Authors: Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)


Abstract
For an n-variate order-d tensor A, define A_{max} := sup_{||x||_2 = 1} <A,x^(otimes d)>, to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d. +1/-1 entries, A_{max} <= sqrt(n.d.log(d)) w.h.p. We study the problem of efficiently certifying upper bounds on A_{max} via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: * When A is a random order-q tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n/q^(1-o(1)))^(q/4-1/2) w.h.p. Our upper bound improves a result of Montanari and Richard (NIPS 2014) when q is large. * We show the above bound is the best possible up to lower order terms, namely the optimum of the level-q SoS relaxation is at least A_{max} * (n/q^(1+o(1)))^(q/4-1/2). * When A is a random order-d tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n*polylog/q)^(d/4 - 1/2) w.h.p. For growing q, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who tightly characterized constant levels of SoS.

Cite as

Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2017.31,
  author =	{Bhattiprolu, Vijay and Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{31:1--31:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.31},
  URN =		{urn:nbn:de:0030-drops-75808},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.31},
  annote =	{Keywords: Sum-of-Squares, Optimization over Sphere, Random Polynomials}
}
Document
Separating a Voronoi Diagram via Local Search

Authors: Vijay V. S. P. Bhattiprolu and Sariel Har-Peled

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)


Abstract
Given a set P of n points in R^d , we show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2 , of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of P , we present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition.

Cite as

Vijay V. S. P. Bhattiprolu and Sariel Har-Peled. Separating a Voronoi Diagram via Local Search. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{bhattiprolu_et_al:LIPIcs.SoCG.2016.18,
  author =	{Bhattiprolu, Vijay V. S. P. and Har-Peled, Sariel},
  title =	{{Separating a Voronoi Diagram via Local Search}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.18},
  URN =		{urn:nbn:de:0030-drops-59107},
  doi =		{10.4230/LIPIcs.SoCG.2016.18},
  annote =	{Keywords: Separators, Local search, Approximation, Voronoi diagrams, Delaunay triangulation, Meshing, Geometric hitting set}
}
Document
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

Authors: Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
A hypergraph is said to be X-colorable if its vertices can be colored with X colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2^(-k+1) of hyperedges (which is trivially achieved by a random 2-coloring), and the best algorithms to color the hypergraph properly require about n^(1-1/k) colors, approaching the trivial bound of n as k increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 2-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 2-colorability: (A) Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l << sqrt(k), we give an algorithm to color the hypergraph with about n^(O(l^2/k)) colors. However, for the maximization version, we prove NP-hardness of finding a 2-coloring miscoloring a smaller than 2^(-O(k)) (resp. k^(-O(k))) fraction of the hyperedges when l = O(log k) (resp. l=2). Assuming the Unique Games conjecture, we improve the latter hardness factor to 2^(-O(k)) for almost discrepancy-1 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k-l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+1)-discrepancy 2-coloring), we give a 2-coloring algorithm that miscolors at most k^(-Omega(k)) of the hyperedges when l << sqrt(k), and complement this with a matching Unique Games hardness result showing that when l = sqrt(k), it is hard to even beat the 2^(-k+1) bound achieved by a random coloring. (C) Strong Colorability: We obtain similar (stronger) Min- and Max-2-Coloring algorithmic results in the case of (k+l)-strong colorability.

Cite as

Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 152-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2015.152,
  author =	{Bhattiprolu, Vijay V. S. P. and Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Approximate Hypergraph Coloring under Low-discrepancy and Related Promises}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{152--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.152},
  URN =		{urn:nbn:de:0030-drops-53011},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.152},
  annote =	{Keywords: Hypergraph Coloring, Discrepancy, Rainbow Coloring, Stong Coloring, Algorithms, Semidefinite Programming, Hardness of Approximation}
}
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