All reports by Author Thomas Watson:

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TR14-055
| 17th April 2014
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Mika Göös, Thomas Watson#### Communication Complexity of Set-Disjointness for All Probabilities

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TR13-124
| 9th September 2013
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Thomas Watson#### The Complexity of Deciding Statistical Properties of Samplable Distributions

Revisions: 1

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TR12-070
| 26th May 2012
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Thomas Watson#### The Complexity of Estimating Min-Entropy

Revisions: 1

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TR12-026
| 27th March 2012
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Thomas Watson#### Time Hierarchies for Sampling Distributions

Revisions: 1

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TR11-120
| 6th September 2011
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Thomas Watson#### Advice Lower Bounds for the Dense Model Theorem

Revisions: 1

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TR11-097
| 7th July 2011
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Thomas Watson#### Lift-and-Project Integrality Gaps for the Traveling Salesperson Problem

Revisions: 1

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TR11-037
| 18th March 2011
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Anindya De, Thomas Watson#### Extractors and Lower Bounds for Locally Samplable Sources

Revisions: 3

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TR10-168
| 9th November 2010
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Thomas Watson#### Pseudorandom Generators for Combinatorial Checkerboards

Revisions: 2

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TR10-147
| 22nd September 2010
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Dieter van Melkebeek, Thomas Watson#### Time-Space Efficient Simulations of Quantum Computations

Revisions: 1

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TR10-126
| 12th August 2010
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Thomas Watson#### Query Complexity in Errorless Hardness Amplification

Revisions: 2

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TR10-042
| 12th March 2010
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Thomas Watson#### Relativized Worlds Without Worst-Case to Average-Case Reductions for NP

Revisions: 3

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TR08-017
| 16th December 2007
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Thomas Watson, Dieter van Melkebeek#### A Quantum Time-Space Lower Bound for the Counting Hierarchy

Mika Göös, Thomas Watson

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions ... more >>>

Thomas Watson

We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all ... more >>>

Thomas Watson

Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBP-complete, even when restricted to 3-local samplers. For logarithmic-space samplers, ... more >>>

Thomas Watson

We prove that for every constant $k\ge 2$, every polynomial time bound $t$, and every polynomially small $\epsilon$, there exists a family of distributions on $k$ elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance $1-1/k-\epsilon$ in time $t$. Our proof involves reducing ... more >>>

Thomas Watson

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution $D$ that is $\delta$-dense in a distribution that is ... more >>>

Thomas Watson

We study the lift-and-project procedures of Lovasz-Schrijver and Sherali-Adams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integrality gap of the standard relaxation is at least 2. We ... more >>>

Anindya De, Thomas Watson

We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number $d$ of the random input bits. As our main result, we construct a deterministic extractor that, given any $d$-local source with ... more >>>

Thomas Watson

We define a combinatorial checkerboard to be a function $f:\{1,\ldots,m\}^d\to\{1,-1\}$ of the form $f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)$ for some functions $f_i:\{1,\ldots,m\}\to\{1,-1\}$. This is a variant of combinatorial rectangles, which can be defined in the same way but using $\{0,1\}$ instead of $\{1,-1\}$. We consider the problem of constructing explicit pseudorandom generators for combinatorial ... more >>>

Dieter van Melkebeek, Thomas Watson

We give two time- and space-efficient simulations of quantum computations with

intermediate measurements, one by classical randomized computations with

unbounded error and the other by quantum computations that use an arbitrary

fixed universal set of gates. Specifically, our simulations show that every

language solvable by a bounded-error quantum algorithm running ...
more >>>

Thomas Watson

An errorless circuit for a boolean function is one that outputs the correct answer or ``don't know'' on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if $f$ has no size $s$ errorless circuit that outputs ``don't know'' on at ... more >>>

Thomas Watson

We prove that relative to an oracle, there is no worst-case to errorless-average-case reduction for $\NP$. This result is the first progress on an open problem posed by Impagliazzo in 1995, namely to construct an oracle relative to which $\NP$ is worst-case hard but errorless-average-case easy. We also handle classes ... more >>>

Thomas Watson, Dieter van Melkebeek

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real $d$ and every positive real $\epsilon$ ... more >>>