16 Search Results for "Chen, Xi"

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
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.39
Subset Sum in Time 2^{n/2} / poly(n)

Authors: Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
A major goal in the area of exact exponential algorithms is to give an algorithm for the (worst-case) n-input Subset Sum problem that runs in time 2^{(1/2 - c)n} for some constant c > 0. In this paper we give a Subset Sum algorithm with worst-case running time O(2^{n/2} ⋅ n^{-γ}) for a constant γ > 0.5023 in standard word RAM or circuit RAM models. To the best of our knowledge, this is the first improvement on the classical "meet-in-the-middle" algorithm for worst-case Subset Sum, due to Horowitz and Sahni, which can be implemented in time O(2^{n/2}) in these memory models [Horowitz and Sahni, 1974]. Our algorithm combines a number of different techniques, including the "representation method" introduced by Howgrave-Graham and Joux [Howgrave-Graham and Joux, 2010] and subsequent adaptations of the method in Austrin, Kaski, Koivisto, and Nederlof [Austrin et al., 2016], and Nederlof and Węgrzycki [Jesper Nederlof and Karol Wegrzycki, 2021], and "bit-packing" techniques used in the work of Baran, Demaine, and Pǎtraşcu [Baran et al., 2005] on subquadratic algorithms for 3SUM.
Cite as

Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio. Subset Sum in Time 2^{n/2} / poly(n). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2023.39,
  author =	{Chen, Xi and Jin, Yaonan and Randolph, Tim and Servedio, Rocco A.},
  title =	{{Subset Sum in Time 2^\{n/2\} / poly(n)}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{39:1--39:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  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.2023.39},
  URN =		{urn:nbn:de:0030-drops-188641},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.39},
  annote =	{Keywords: Exact algorithms, subset sum, log shaving}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.21
Reducing Tarski to Unique Tarski (In the Black-Box Model)

Authors: Xi Chen, Yuhao Li, and Mihalis Yannakakis

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

Abstract
We study the problem of finding a Tarski fixed point over the k-dimensional grid [n]^k. We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point.
Cite as

Xi Chen, Yuhao Li, and Mihalis Yannakakis. Reducing Tarski to Unique Tarski (In the Black-Box Model). In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chen_et_al:LIPIcs.CCC.2023.21,
  author =	{Chen, Xi and Li, Yuhao and Yannakakis, Mihalis},
  title =	{{Reducing Tarski to Unique Tarski (In the Black-Box Model)}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{21:1--21:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.21},
  URN =		{urn:nbn:de:0030-drops-182919},
  doi =		{10.4230/LIPIcs.CCC.2023.21},
  annote =	{Keywords: Tarski fixed point, Query complexity, TFNP}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.75
Learning Reserve Prices in Second-Price Auctions

Authors: Yaonan Jin, Pinyan Lu, and Tao Xiao

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

Abstract
This paper proves the tight sample complexity of Second-Price Auction with Anonymous Reserve, up to a logarithmic factor, for each of all the value distribution families studied in the literature: [0,1]-bounded, [1,H]-bounded, regular, and monotone hazard rate (MHR). Remarkably, the setting-specific tight sample complexity poly(ε^{-1}) depends on the precision ε ∈ (0, 1), but not on the number of bidders n ≥ 1. Further, in the two bounded-support settings, our learning algorithm allows correlated value distributions. In contrast, the tight sample complexity Θ̃(n) ⋅ poly(ε^{-1}) of Myerson Auction proved by Guo, Huang and Zhang (STOC 2019) has a nearly-linear dependence on n ≥ 1, and holds only for independent value distributions in every setting. We follow a similar framework as the Guo-Huang-Zhang work, but replace their information theoretical arguments with a direct proof.
Cite as

Yaonan Jin, Pinyan Lu, and Tao Xiao. Learning Reserve Prices in Second-Price Auctions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 75:1-75:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jin_et_al:LIPIcs.ITCS.2023.75,
  author =	{Jin, Yaonan and Lu, Pinyan and Xiao, Tao},
  title =	{{Learning Reserve Prices in Second-Price Auctions}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{75:1--75:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.75},
  URN =		{urn:nbn:de:0030-drops-175780},
  doi =		{10.4230/LIPIcs.ITCS.2023.75},
  annote =	{Keywords: Revenue Maximization, Sample Complexity, Anonymous Reserve}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.19
The Composition Complexity of Majority

Authors: Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan

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

Abstract
We study the complexity of computing majority as a composition of local functions: Maj_n = h(g_1,…,g_m), where each g_j: {0,1}ⁿ → {0,1} is an arbitrary function that queries only k ≪ n variables and h: {0,1}^m → {0,1} is an arbitrary combining function. We prove an optimal lower bound of m ≥ Ω(n/k log k) on the number of functions needed, which is a factor Ω(log k) larger than the ideal m = n/k. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority. Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions g_j, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).
Cite as

Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan. The Composition Complexity of Majority. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 19:1-19:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lecomte_et_al:LIPIcs.CCC.2022.19,
  author =	{Lecomte, Victor and Ramakrishnan, Prasanna and Tan, Li-Yang},
  title =	{{The Composition Complexity of Majority}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{19:1--19:26},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.19},
  URN =		{urn:nbn:de:0030-drops-165818},
  doi =		{10.4230/LIPIcs.CCC.2022.19},
  annote =	{Keywords: computational complexity, circuit lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.20
Polynomial-Time Trace Reconstruction in the Low Deletion Rate Regime

Authors: Xi Chen, Anindya De, Chin Ho Lee, Rocco A. Servedio, and Sandip Sinha

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

Abstract
In the trace reconstruction problem, an unknown source string x ∈ {0,1}ⁿ is transmitted through a probabilistic deletion channel which independently deletes each bit with some fixed probability δ and concatenates the surviving bits, resulting in a trace of x. The problem is to reconstruct x given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly(n)-time algorithms being the 2004 algorithm of Batu et al. [T. Batu et al., 2004]. This algorithm can reconstruct an arbitrary source string x ∈ {0,1}ⁿ in poly(n) time provided that the deletion rate δ satisfies δ ≤ n^{-(1/2 + ε)} for some ε > 0. In this work we improve on the result of [T. Batu et al., 2004] by giving a poly(n)-time algorithm for trace reconstruction for any deletion rate δ ≤ n^{-(1/3 + ε)}. Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.
Cite as

Xi Chen, Anindya De, Chin Ho Lee, Rocco A. Servedio, and Sandip Sinha. Polynomial-Time Trace Reconstruction in the Low Deletion Rate Regime. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2021.20,
  author =	{Chen, Xi and De, Anindya and Lee, Chin Ho and Servedio, Rocco A. and Sinha, Sandip},
  title =	{{Polynomial-Time Trace Reconstruction in the Low Deletion Rate Regime}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{20:1--20:20},
  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.2021.20},
  URN =		{urn:nbn:de:0030-drops-135595},
  doi =		{10.4230/LIPIcs.ITCS.2021.20},
  annote =	{Keywords: trace reconstruction}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.44
Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions

Authors: Frank Ban, Xi Chen, Rocco A. Servedio, and Sandip Sinha

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
A number of recent works have considered the trace reconstruction problem, in which an unknown source string x in {0,1}^n is transmitted through a probabilistic channel which may randomly delete coordinates or insert random bits, resulting in a trace of x. The goal is to reconstruct the original string x from independent traces of x. While the asymptotically best algorithms known for worst-case strings use exp(O(n^{1/3})) traces [De et al., 2017; Fedor Nazarov and Yuval Peres, 2017], several highly efficient algorithms are known [Yuval Peres and Alex Zhai, 2017; Nina Holden et al., 2018] for the average-case version of the problem, in which the source string x is chosen uniformly at random from {0,1}^n. In this paper we consider a generalization of the above-described average-case trace reconstruction problem, which we call average-case population recovery in the presence of insertions and deletions. In this problem, rather than a single unknown source string there is an unknown distribution over s unknown source strings x^1,...,x^s in {0,1}^n, and each sample given to the algorithm is independently generated by drawing some x^i from this distribution and returning an independent trace of x^i. Building on the results of [Yuval Peres and Alex Zhai, 2017] and [Nina Holden et al., 2018], we give an efficient algorithm for the average-case population recovery problem in the presence of insertions and deletions. For any support size 1 <= s <= exp(Theta(n^{1/3})), for a 1-o(1) fraction of all s-element support sets {x^1,...,x^s} subset {0,1}^n, for every distribution D supported on {x^1,...,x^s}, our algorithm can efficiently recover D up to total variation distance at most epsilon with high probability, given access to independent traces of independent draws from D. The running time of our algorithm is poly(n,s,1/epsilon) and its sample complexity is poly (s,1/epsilon,exp(log^{1/3} n)). This polynomial dependence on the support size s is in sharp contrast with the worst-case version of the problem (when x^1,...,x^s may be any strings in {0,1}^n), in which the sample complexity of the most efficient known algorithm [Frank Ban et al., 2019] is doubly exponential in s.
Cite as

Frank Ban, Xi Chen, Rocco A. Servedio, and Sandip Sinha. Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ban_et_al:LIPIcs.APPROX-RANDOM.2019.44,
  author =	{Ban, Frank and Chen, Xi and Servedio, Rocco A. and Sinha, Sandip},
  title =	{{Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{44:1--44:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  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.2019.44},
  URN =		{urn:nbn:de:0030-drops-112592},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.44},
  annote =	{Keywords: population recovery, deletion channel, trace reconstruction}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.2
Almost Optimal Distribution-Free Junta Testing

Authors: Nader H. Bshouty

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries.
Cite as

Nader H. Bshouty. Almost Optimal Distribution-Free Junta Testing. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bshouty:LIPIcs.CCC.2019.2,
  author =	{Bshouty, Nader H.},
  title =	{{Almost Optimal Distribution-Free Junta Testing}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{2:1--2:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.2},
  URN =		{urn:nbn:de:0030-drops-108249},
  doi =		{10.4230/LIPIcs.CCC.2019.2},
  annote =	{Keywords: Distribution-free property testing, k-Junta}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.4
Simple and Efficient Pseudorandom Generators from Gaussian Processes

Authors: Eshan Chattopadhyay, Anindya De, and Rocco A. Servedio

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
We show that a very simple pseudorandom generator fools intersections of k linear threshold functions (LTFs) and arbitrary functions of k LTFs over n-dimensional Gaussian space. The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of PRGs for functions of halfspaces. Our analysis for arbitrary functions of LTFs establishes bounds on the Wasserstein distance between Gaussian random vectors with similar covariance matrices, and combines these bounds with a conversion from Wasserstein distance to "union-of-orthants" distance from [Xi Chen et al., 2014]. Our analysis for intersections of LTFs uses extensions of the classical Sudakov-Fernique type inequalities, which give bounds on the difference between the expectations of the maxima of two Gaussian random vectors with similar covariance matrices. For all values of k, our generator has seed length O(log n) + poly(k) for arbitrary functions of k LTFs and O(log n) + poly(log k) for intersections of k LTFs. The best previous result, due to [Gopalan et al., 2010], only gave such PRGs for arbitrary functions of k LTFs when k=O(log log n) and for intersections of k LTFs when k=O((log n)/(log log n)). Thus our PRG achieves an O(log n) seed length for values of k that are exponentially larger than previous work could achieve. By combining our PRG over Gaussian space with an invariance principle for arbitrary functions of LTFs and with a regularity lemma, we obtain a deterministic algorithm that approximately counts satisfying assignments of arbitrary functions of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(k,1/epsilon)} for all values of k. This algorithm has a poly(n) runtime for k =(log n)^c for some absolute constant c>0, while the previous best poly(n)-time algorithms could only handle k = O(log log n). For intersections of LTFs, by combining these tools with a recent PRG due to [R. O'Donnell et al., 2018], we obtain a deterministic algorithm that can approximately count satisfying assignments of intersections of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(log k, 1/epsilon)}. This algorithm has a poly(n) runtime for k =2^{(log n)^c} for some absolute constant c>0, while the previous best poly(n)-time algorithms for intersections of k LTFs, due to [Gopalan et al., 2010], could only handle k=O((log n)/(log log n)).
Cite as

Eshan Chattopadhyay, Anindya De, and Rocco A. Servedio. Simple and Efficient Pseudorandom Generators from Gaussian Processes. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 4:1-4:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2019.4,
  author =	{Chattopadhyay, Eshan and De, Anindya and Servedio, Rocco A.},
  title =	{{Simple and Efficient Pseudorandom Generators from Gaussian Processes}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{4:1--4:33},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.4},
  URN =		{urn:nbn:de:0030-drops-108262},
  doi =		{10.4230/LIPIcs.CCC.2019.4},
  annote =	{Keywords: Polynomial threshold functions, Gaussian processes, Johnson-Lindenstrauss, pseudorandom generators}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.52
Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

Authors: Amit Levi and Erik Waingarten

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}: - Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries. - Tolerant unateness testing requires Omega~(n) queries. - Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries. Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.
Cite as

Amit Levi and Erik Waingarten. Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{levi_et_al:LIPIcs.ITCS.2019.52,
  author =	{Levi, Amit and Waingarten, Erik},
  title =	{{Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{52:1--52:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.52},
  URN =		{urn:nbn:de:0030-drops-101452},
  doi =		{10.4230/LIPIcs.ITCS.2019.52},
  annote =	{Keywords: Property Testing, Juntas, Tolerant Testing, Boolean functions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.57
Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games

Authors: Xi Chen, Yu Cheng, and Bo Tang

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
In this paper we present a generic reduction from the problem of finding an \epsilon-well-supported Nash equilibrium (WSNE) to that of finding an \Theta(\epsilon)-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player. Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE. This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions. As an application we prove a 2^{\Omega(n/\log n)} lower bound on the randomized query complexity of finding an \epsilon-ANE in binary-action n-player games, for some constant \epsilon>0. This answers an open problem posed by Hart and Nisan and Babichenko, and is very close to the trivial upper bound of 2^n. Previously for WSNE, Babichenko showed a 2^{\Omega(n)} lower bound on the randomized query complexity of finding an \epsilon-WSNE for some constant epsilon>0. Our result follows directly from combining Babichenko's result and our new reduction from WSNE to ANE.
Cite as

Xi Chen, Yu Cheng, and Bo Tang. Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 57:1-57:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2017.57,
  author =	{Chen, Xi and Cheng, Yu and Tang, Bo},
  title =	{{Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{57:1--57:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.57},
  URN =		{urn:nbn:de:0030-drops-81636},
  doi =		{10.4230/LIPIcs.ITCS.2017.57},
  annote =	{Keywords: Equilibrium Computation, Query Complexity, Large Games}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.37
Sample-Based High-Dimensional Convexity Testing

Authors: Xi Chen, Adam Freilich, Rocco A. Servedio, and Timothy Sun

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

Abstract
In the problem of high-dimensional convexity testing, there is an unknown set S in the n-dimensional Euclidean space which is promised to be either convex or c-far from every convex body with respect to the standard multivariate normal distribution. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set S as possible. In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples (x,S(x)) where each x is independently drawn from the normal distribution. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant c, our results show that the sample complexity of one-sided convexity testing is exponential in n, while for two-sided convexity testing it is exponential in the square root of n.
Cite as

Xi Chen, Adam Freilich, Rocco A. Servedio, and Timothy Sun. Sample-Based High-Dimensional Convexity Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2017.37,
  author =	{Chen, Xi and Freilich, Adam and Servedio, Rocco A. and Sun, Timothy},
  title =	{{Sample-Based High-Dimensional Convexity Testing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{37:1--37:20},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.37},
  URN =		{urn:nbn:de:0030-drops-75867},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.37},
  annote =	{Keywords: Property testing, convexity, sample-based testing}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.38
Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces

Authors: Xi Chen, Rocco A. Servedio, Li-Yang Tan, and Erik Waingarten

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

Abstract
We give a poly(log(n),1/epsilon)-query adaptive algorithm for testing whether an unknown Boolean function f:{-1, 1}^n -> {-1, 1}, which is promised to be a halfspace, is monotone versus epsilon-far from monotone. Since non-adaptive algorithms are known to require almost Omega(n^{1/2}) queries to test whether an unknown halfspace is monotone versus far from monotone, this shows that adaptivity enables an exponential improvement in the query complexity of monotonicity testing for halfspaces.
Cite as

Xi Chen, Rocco A. Servedio, Li-Yang Tan, and Erik Waingarten. Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2017.38,
  author =	{Chen, Xi and Servedio, Rocco A. and Tan, Li-Yang and Waingarten, Erik},
  title =	{{Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{38:1--38:21},
  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.2017.38},
  URN =		{urn:nbn:de:0030-drops-75877},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.38},
  annote =	{Keywords: property testing, linear threshold functions, monotonicity, adaptivity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.26
Settling the Query Complexity of Non-Adaptive Junta Testing

Authors: Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, and Jinyu Xie

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f is a k-junta or epsilon-far from every k-junta must make ~Omega(k^{3/2}/ epsilon) many queries for a wide range of parameters k and epsilon. Our result dramatically improves previous lower bounds from [BGSMdW13,STW15], and is essentially optimal given Blais's non-adaptive junta tester from [Blais08], which makes ~O(k^{3/2})/epsilon queries. Combined with the adaptive tester of [Blais09] which makes O(k log k + k / epsilon) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing.
Cite as

Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, and Jinyu Xie. Settling the Query Complexity of Non-Adaptive Junta Testing. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.CCC.2017.26,
  author =	{Chen, Xi and Servedio, Rocco A. and Tan, Li-Yang and Waingarten, Erik and Xie, Jinyu},
  title =	{{Settling the Query Complexity of Non-Adaptive Junta Testing}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.26},
  URN =		{urn:nbn:de:0030-drops-75283},
  doi =		{10.4230/LIPIcs.CCC.2017.26},
  annote =	{Keywords: property testing, juntas, query complexity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.27
Some Lower Bounds in Parameterized AC^0

Authors: Yijia Chen and Jörg Flum

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical AC^0. Among others, we derive such a lower bound for all fpt-approximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong AC^0 version of the planted clique conjecture: AC^0-circuits asymptotically almost surely can not distinguish between a random graph and this graph with a randomly planted clique of any size <= n^xi (where 0 <= xi < 1).
Cite as

Yijia Chen and Jörg Flum. Some Lower Bounds in Parameterized AC^0. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chen_et_al:LIPIcs.MFCS.2016.27,
  author =	{Chen, Yijia and Flum, J\"{o}rg},
  title =	{{Some Lower Bounds in Parameterized AC^0}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.27},
  URN =		{urn:nbn:de:0030-drops-64423},
  doi =		{10.4230/LIPIcs.MFCS.2016.27},
  annote =	{Keywords: parameterized AC^0, lower bound, clique, halting problem}
}
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