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Documents authored by Nadimpalli, Shivam

Document
DOI: 10.4230/LIPIcs.ITCS.2024.33
Testing Intersecting and Union-Closed Families

Authors: Xi Chen, Anindya De, Yuhao Li, Shivam Nadimpalli, and Rocco A. Servedio

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically intersecting families and union-closed families. A function f: {0,1}ⁿ → {0,1} is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of [n]. Our main results are that - in sharp contrast with the property of being a monotone set system - the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: - For ε ≥ Ω(1/√n), any non-adaptive two-sided ε-tester for intersectingness must make 2^{Ω(n^{1/4}/√{ε})} queries. We also give a 2^{Ω(√{n log(1/ε)})}-query lower bound for non-adaptive one-sided ε-testers for intersectingness. - For ε ≥ 1/2^{Ω(n^{0.49})}, any non-adaptive two-sided ε-tester for union-closedness must make n^{Ω(log(1/ε))} queries. Thus, neither intersectingness nor union-closedness shares the poly(n,1/ε)-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple poly(n^{√{nlog(1/ε)}},1/ε)-query, one-sided, non-adaptive algorithm for ε-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when ε = Θ(1/√n), and for one-sided testing of intersectingness when ε = Θ(1).
Cite as

Xi Chen, Anindya De, Yuhao Li, Shivam Nadimpalli, and Rocco A. Servedio. Testing Intersecting and Union-Closed Families. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2024.33,
  author =	{Chen, Xi and De, Anindya and Li, Yuhao and Nadimpalli, Shivam and Servedio, Rocco A.},
  title =	{{Testing Intersecting and Union-Closed Families}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{33:1--33:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.33},
  URN =		{urn:nbn:de:0030-drops-195610},
  doi =		{10.4230/LIPIcs.ITCS.2024.33},
  annote =	{Keywords: Sublinear algorithms, property testing, computational complexity, monotonicity, intersecting families, union-closed families}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.53
Convex Influences

Authors: Anindya De, Shivam Nadimpalli, and Rocco A. Servedio

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

Abstract
We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions f: {±1}ⁿ → {±1}. Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincaré inequality, the Kahn-Kalai-Linial theorem [J. Kahn et al., 1988], a sharp threshold theorem of Kalai [G. Kalai, 2004], a stability version of the Kruskal-Katona theorem due to O'Donnell and Wimmer [R. O'Donnell and K. Wimmer, 2009], and some partial results towards a Gaussian space analogue of Friedgut’s junta theorem [E. Friedgut, 1998]. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over {±1}ⁿ. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from {±1}ⁿ to {±1}.
Cite as

Anindya De, Shivam Nadimpalli, and Rocco A. Servedio. Convex Influences. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{de_et_al:LIPIcs.ITCS.2022.53,
  author =	{De, Anindya and Nadimpalli, Shivam and Servedio, Rocco A.},
  title =	{{Convex Influences}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{53:1--53:21},
  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.2022.53},
  URN =		{urn:nbn:de:0030-drops-156498},
  doi =		{10.4230/LIPIcs.ITCS.2022.53},
  annote =	{Keywords: Fourier analysis of Boolean functions, convex geometry, influences, threshold phenomena}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.69
Quantitative Correlation Inequalities via Semigroup Interpolation

Authors: Anindya De, Shivam Nadimpalli, and Rocco A. Servedio

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

Abstract
Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. We give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic analysis of functions over product spaces. We instantiate this general approach in several different concrete settings to obtain a range of new and near-optimal quantitative correlation inequalities, including: - A {quantitative} version of Royen’s celebrated Gaussian Correlation Inequality [Royen, 2014]. In [Royen, 2014] Royen confirmed a conjecture, open for 40 years, stating that any two symmetric convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, conceptually similar to Talagrand’s quantitative correlation bound for monotone Boolean functions over {0,1}ⁿ [M. Talagrand, 1996]. We show that our quantitative version of Royen’s theorem is within a logarithmic factor of being optimal. - A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space. This is a broad generalization of Talagrand’s quantitative correlation bound for functions from {0,1}ⁿ to {0,1} under the uniform distribution [M. Talagrand, 1996]; the only prior generalization of which we are aware is due to Keller [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009], which extended [M. Talagrand, 1996] to product distributions over {0,1}ⁿ. In the special case of p-biased distributions over {0,1}ⁿ that was considered by Keller, our new bound essentially saves a factor of p log(1/p) over the quantitative bounds given in [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009]. We also give {a quantitative version of} the FKG inequality for monotone functions over the continuous domain [0,1]ⁿ, answering a question of Keller [Nathan Keller, 2009].
Cite as

Anindya De, Shivam Nadimpalli, and Rocco A. Servedio. Quantitative Correlation Inequalities via Semigroup Interpolation. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 69:1-69:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{de_et_al:LIPIcs.ITCS.2021.69,
  author =	{De, Anindya and Nadimpalli, Shivam and Servedio, Rocco A.},
  title =	{{Quantitative Correlation Inequalities via Semigroup Interpolation}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{69:1--69:20},
  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.2021.69},
  URN =		{urn:nbn:de:0030-drops-136081},
  doi =		{10.4230/LIPIcs.ITCS.2021.69},
  annote =	{Keywords: complex analysis, correlation inequality, FKG inequality, Gaussian correlation inequality, harmonic analysis, Markov semigroups}
}
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