24 Search Results for "Watson, Thomas"

Document
DOI: 10.4230/LIPIcs.TQC.2023.3
Optimal Algorithms for Learning Quantum Phase States

Authors: Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract
We analyze the complexity of learning n-qubit quantum phase states. A degree-d phase state is defined as a superposition of all 2ⁿ basis vectors x with amplitudes proportional to (-1)^{f(x)}, where f is a degree-d Boolean polynomial over n variables. We show that the sample complexity of learning an unknown degree-d phase state is Θ(n^d) if we allow separable measurements and Θ(n^{d-1}) if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime poly(n) (for constant d) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli X and Z bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when f has sparsity-s, degree-d in its 𝔽₂ representation (with sample complexity O(2^d sn)), f has Fourier-degree-t (with sample complexity O(2^{2t})), and learning quadratic phase states with ε-global depolarizing noise (with sample complexity O(n^{1+ε})). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP circuits.
Cite as

Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder. Optimal Algorithms for Learning Quantum Phase States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 3:1-3:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arunachalam_et_al:LIPIcs.TQC.2023.3,
  author =	{Arunachalam, Srinivasan and Bravyi, Sergey and Dutt, Arkopal and Yoder, Theodore J.},
  title =	{{Optimal Algorithms for Learning Quantum Phase States}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{3:1--3:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.3},
  URN =		{urn:nbn:de:0030-drops-183139},
  doi =		{10.4230/LIPIcs.TQC.2023.3},
  annote =	{Keywords: Tomography, binary phase states, generalized phase states, IQP circuits}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.26
Complexity of Fault Tolerant Query Complexity

Authors: Ramita Maharjan and Thomas Watson

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
In the model of fault tolerant decision trees introduced by Kenyon and Yao, there is a known upper bound E on the total number of queries that may be faulty (i.e., get the wrong bit). We consider this computational problem: Given as input the truth table of a function f: {0,1}ⁿ → {0,1} and a value of E, find the minimum possible height (worst-case number of queries) of any decision tree that computes f while tolerating up to E many faults. We design an algorithm for this problem that runs in time Õ(binom(n+E,E)⋅(2E+3)ⁿ), which is polynomial in the size of the truth table when E is a constant. This generalizes a standard algorithm for the non-fault tolerant setting.
Cite as

Ramita Maharjan and Thomas Watson. Complexity of Fault Tolerant Query Complexity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{maharjan_et_al:LIPIcs.FSTTCS.2022.26,
  author =	{Maharjan, Ramita and Watson, Thomas},
  title =	{{Complexity of Fault Tolerant Query Complexity}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{26:1--26:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.26},
  URN =		{urn:nbn:de:0030-drops-174185},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.26},
  annote =	{Keywords: Fault, Tolerant, Query, Complexity}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.3
The Parametrized Complexity of Quantum Verification

Authors: Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman

Published in: LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Abstract
We initiate the study of parameterized complexity of QMA problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + t T-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most t qubits (independent of the system size). Furthermore, we derive new lower bounds on the T-count of circuit satisfiability instances and the T-count of the W-state assuming the classical exponential time hypothesis (ETH). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.
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Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman. The Parametrized Complexity of Quantum Verification. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{arunachalam_et_al:LIPIcs.TQC.2022.3,
  author =	{Arunachalam, Srinivasan and Bravyi, Sergey and Nirkhe, Chinmay and O'Gorman, Bryan},
  title =	{{The Parametrized Complexity of Quantum Verification}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{3:1--3:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-237-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{232},
  editor =	{Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.3},
  URN =		{urn:nbn:de:0030-drops-165104},
  doi =		{10.4230/LIPIcs.TQC.2022.3},
  annote =	{Keywords: parametrized complexity, quantum verification, QMA}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.31
Erdős-Selfridge Theorem for Nonmonotone CNFs

Authors: Md Lutfar Rahman and Thomas Watson

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
In an influential paper, Erdős and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker’s goal is to satisfy the CNF, while Maker’s goal is to falsify it. The Erdős-Selfridge Theorem says that the least number of clauses in any monotone CNF with k literals per clause where Maker has a winning strategy is Θ(2^k). We study the analogous question when the CNF is not necessarily monotone. We prove bounds of Θ(√2 ^k) when Maker plays last, and Ω(1.5^k) and O(r^k) when Breaker plays last, where r = (1+√5)/2≈ 1.618 is the golden ratio.
Cite as

Md Lutfar Rahman and Thomas Watson. Erdős-Selfridge Theorem for Nonmonotone CNFs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 31:1-31:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rahman_et_al:LIPIcs.SWAT.2022.31,
  author =	{Rahman, Md Lutfar and Watson, Thomas},
  title =	{{Erd\H{o}s-Selfridge Theorem for Nonmonotone CNFs}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{31:1--31:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.31},
  URN =		{urn:nbn:de:0030-drops-161916},
  doi =		{10.4230/LIPIcs.SWAT.2022.31},
  annote =	{Keywords: Game, nonmonotone, CNFs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.18
A Majority Lemma for Randomised Query Complexity

Authors: Mika Göös and Gilbert Maystre

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
We show that computing the majority of n copies of a boolean function g has randomised query complexity R(Maj∘gⁿ) = Θ(n⋅R ̅_{1/n}(g)). In fact, we show that to obtain a similar result for any composed function f∘gⁿ, it suffices to prove a sufficiently strong form of the result only in the special case g = GapOr.
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Mika Göös and Gilbert Maystre. A Majority Lemma for Randomised Query Complexity. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goos_et_al:LIPIcs.CCC.2021.18,
  author =	{G\"{o}\"{o}s, Mika and Maystre, Gilbert},
  title =	{{A Majority Lemma for Randomised Query Complexity}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.18},
  URN =		{urn:nbn:de:0030-drops-142922},
  doi =		{10.4230/LIPIcs.CCC.2021.18},
  annote =	{Keywords: Query Complexity, Composition, Majority}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.57
6-Uniform Maker-Breaker Game Is PSPACE-Complete

Authors: Md Lutfar Rahman and Thomas Watson

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
In a STOC 1976 paper, Schaefer proved that it is PSPACE-complete to determine the winner of the so-called Maker-Breaker game on a given set system, even when every set has size at most 11. Since then, there has been no improvement on this result. We prove that the game remains PSPACE-complete even when every set has size 6.
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Md Lutfar Rahman and Thomas Watson. 6-Uniform Maker-Breaker Game Is PSPACE-Complete. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{rahman_et_al:LIPIcs.STACS.2021.57,
  author =	{Rahman, Md Lutfar and Watson, Thomas},
  title =	{{6-Uniform Maker-Breaker Game Is PSPACE-Complete}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.57},
  URN =		{urn:nbn:de:0030-drops-137020},
  doi =		{10.4230/LIPIcs.STACS.2021.57},
  annote =	{Keywords: Game, Maker-Breaker, Complexity, Reduction, PSPACE-complete, NL-hard}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.28
When Is Amplification Necessary for Composition in Randomized Query Complexity?

Authors: Shalev Ben-David, Mika Göös, Robin Kothari, and Thomas Watson

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

Abstract
Suppose we have randomized decision trees for an outer function f and an inner function g. The natural approach for obtaining a randomized decision tree for the composed function (f∘ gⁿ)(x¹,…,xⁿ) = f(g(x¹),…,g(xⁿ)) involves amplifying the success probability of the decision tree for g, so that a union bound can be used to bound the error probability over all the coordinates. The amplification introduces a logarithmic factor cost overhead. We study the question: When is this log factor necessary? We show that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Our results are related to, but qualitatively strengthen in various ways, known results about decision trees with noisy inputs.
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Shalev Ben-David, Mika Göös, Robin Kothari, and Thomas Watson. When Is Amplification Necessary for Composition in Randomized Query Complexity?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bendavid_et_al:LIPIcs.APPROX/RANDOM.2020.28,
  author =	{Ben-David, Shalev and G\"{o}\"{o}s, Mika and Kothari, Robin and Watson, Thomas},
  title =	{{When Is Amplification Necessary for Composition in Randomized Query Complexity?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{28:1--28:16},
  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.2020.28},
  URN =		{urn:nbn:de:0030-drops-126316},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.28},
  annote =	{Keywords: Amplification, composition, query complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.7
Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

Authors: Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = Ω(|S|) or T = Ω(√{N/|S|}). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA. In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state |S⟩ = 1/√|S| ∑_{i ∈ S} |i⟩ - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either Θ(√{N/|S|}) queries or else Θ(min{|S|^{1/3},√{N/|S|}}) QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.
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Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 7:1-7:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2020.7,
  author =	{Aaronson, Scott and Kothari, Robin and Kretschmer, William and Thaler, Justin},
  title =	{{Quantum Lower Bounds for Approximate Counting via Laurent Polynomials}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{7:1--7:47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.7},
  URN =		{urn:nbn:de:0030-drops-125593},
  doi =		{10.4230/LIPIcs.CCC.2020.7},
  annote =	{Keywords: Approximate counting, Laurent polynomials, QSampling, query complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.92
Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

Authors: Toniann Pitassi, Morgan Shirley, and Thomas Watson

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function with an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sided-error randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.
Cite as

Toniann Pitassi, Morgan Shirley, and Thomas Watson. Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 92:1-92:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pitassi_et_al:LIPIcs.ICALP.2020.92,
  author =	{Pitassi, Toniann and Shirley, Morgan and Watson, Thomas},
  title =	{{Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{92:1--92:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.92},
  URN =		{urn:nbn:de:0030-drops-124992},
  doi =		{10.4230/LIPIcs.ICALP.2020.92},
  annote =	{Keywords: Boolean hierarchies, lifting theorems, query complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.28
Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples

Authors: Ronitt Rubinfeld and Arsen Vasilyan

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.
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Ronitt Rubinfeld and Arsen Vasilyan. Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 28:1-28:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{rubinfeld_et_al:LIPIcs.ITCS.2020.28,
  author =	{Rubinfeld, Ronitt and Vasilyan, Arsen},
  title =	{{Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{28:1--28:34},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.28},
  URN =		{urn:nbn:de:0030-drops-117138},
  doi =		{10.4230/LIPIcs.ITCS.2020.28},
  annote =	{Keywords: Learning distributions, monotone probability distributions, estimating support size}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.69
Testing Properties of Multiple Distributions with Few Samples

Authors: Maryam Aliakbarpour and Sandeep Silwal

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far from q in ℓ_1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.
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Maryam Aliakbarpour and Sandeep Silwal. Testing Properties of Multiple Distributions with Few Samples. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 69:1-69:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aliakbarpour_et_al:LIPIcs.ITCS.2020.69,
  author =	{Aliakbarpour, Maryam and Silwal, Sandeep},
  title =	{{Testing Properties of Multiple Distributions with Few Samples}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{69:1--69:41},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.69},
  URN =		{urn:nbn:de:0030-drops-117545},
  doi =		{10.4230/LIPIcs.ITCS.2020.69},
  annote =	{Keywords: Hypothesis Testing, Property Testing, Distribution Testing, Identity Testing, Closeness Testing, Multiple Sources}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.51
A Lower Bound for Sampling Disjoint Sets

Authors: Mika Göös and Thomas Watson

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

Abstract
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta<1, this requires Omega(n) communication even to get within statistical distance 1-beta^n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Omega(sqrt{n}) communication is required to get within some constant statistical distance epsilon>0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}.
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Mika Göös and Thomas Watson. A Lower Bound for Sampling Disjoint Sets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goos_et_al:LIPIcs.APPROX-RANDOM.2019.51,
  author =	{G\"{o}\"{o}s, Mika and Watson, Thomas},
  title =	{{A Lower Bound for Sampling Disjoint Sets}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{51:1--51:13},
  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.2019.51},
  URN =		{urn:nbn:de:0030-drops-112666},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.51},
  annote =	{Keywords: Communication complexity, set disjointness, sampling}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.96
Amplification with One NP Oracle Query

Authors: Thomas Watson

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
We provide a complete picture of the extent to which amplification of success probability is possible for randomized algorithms having access to one NP oracle query, in the settings of two-sided, one-sided, and zero-sided error. We generalize this picture to amplifying one-query algorithms with q-query algorithms, and we show our inclusions are tight for relativizing techniques.
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Thomas Watson. Amplification with One NP Oracle Query. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 96:1-96:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{watson:LIPIcs.ICALP.2019.96,
  author =	{Watson, Thomas},
  title =	{{Amplification with One NP Oracle Query}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{96:1--96:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.96},
  URN =		{urn:nbn:de:0030-drops-106726},
  doi =		{10.4230/LIPIcs.ICALP.2019.96},
  annote =	{Keywords: Amplification, NP, oracle, query}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.59
A ZPP^NP[1] Lifting Theorem

Authors: Thomas Watson

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
The complexity class ZPP^{NP[1]} (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. - For starters, we provide a new characterization: ZPP^{NP[1]} equals the restriction of BPP^{NP[1]} where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. - Using the above characterization, we prove a query-to-communication lifting theorem, which translates any ZPP^{NP[1]} decision tree lower bound for a function f into a ZPP^{NP[1]} communication lower bound for a two-party version of f. - As an application, we use the above lifting theorem to prove that the ZPP^{NP[1]} communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a "primal" characterization of this lower bound technique as a complexity class.
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Thomas Watson. A ZPP^NP[1] Lifting Theorem. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 59:1-59:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{watson:LIPIcs.STACS.2019.59,
  author =	{Watson, Thomas},
  title =	{{A ZPP^NP\lbrack1\rbrack Lifting Theorem}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{59:1--59:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.59},
  URN =		{urn:nbn:de:0030-drops-102989},
  doi =		{10.4230/LIPIcs.STACS.2019.59},
  annote =	{Keywords: Query complexity, communication complexity, lifting}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.9
Complexity of Unordered CNF Games

Authors: Md Lutfar Rahman and Thomas Watson

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
The classic TQBF problem is to determine who has a winning strategy in a game played on a given CNF formula, where the two players alternate turns picking truth values for the variables in a given order, and the winner is determined by whether the CNF gets satisfied. We study variants of this game in which the variables may be played in any order, and each turn consists of picking a remaining variable and a truth value for it. - For the version where the set of variables is partitioned into two halves and each player may only pick variables from his/her half, we prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for unbounded-width CNFs (Schaefer, STOC 1976). - For the general unordered version (where each variable can be picked by either player), we also prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for 6-CNFs (Ahlroth and Orponen, MFCS 2012) and PSPACE-complete for positive 11-CNFs (Schaefer, STOC 1976).
Cite as

Md Lutfar Rahman and Thomas Watson. Complexity of Unordered CNF Games. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{rahman_et_al:LIPIcs.ISAAC.2018.9,
  author =	{Rahman, Md Lutfar and Watson, Thomas},
  title =	{{Complexity of Unordered CNF Games}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{9:1--9:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.9},
  URN =		{urn:nbn:de:0030-drops-99574},
  doi =		{10.4230/LIPIcs.ISAAC.2018.9},
  annote =	{Keywords: CNF, Games, PSPACE-complete, SAT, Linear Time}
}
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