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Every randomized algorithm can be simulated by an deterministic algorithm, at the expense of an exponential increase in running time. Derandomization is about converting randomized algorithms into efficient deterministic agorithms.

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Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)

Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
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1answer
197 views

Distributions which are intractable to sample from?

I'm looking for an interesting family of probability distributions $P$ that is intractable to efficiently sample from. I'm not sure what the right notion of intractable is, though I know the notion ...
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1answer
84 views

$BPL$ with polylog random bits is in $L$

Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$. My question is ...
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59 views

Applications for non-commutative Khinchine inequality

I am looking for the applications of non-commutative Khinchine inequality (see below) in case when Rademacher random variables are tight by the condition $\sum_{i=1}^N\varepsilon_i=M, \, -N \leq M\leq ...
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0answers
88 views

Does $P=BPP$ say anything about space complexity?

There are many streaming algorithms with sublinear randomized space but linear deterministic space. Does $P=BPP$ have anything to do with derandomizing space and more importantly but not related to ...
5
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1answer
200 views

From $PIT\in P$ to $P=BPP$

If $PIT$ has been derandomized then still how far would we be from showing $P=BPP$? What additional barriers need to be climbed?
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54 views

Does deterministic PIT produce deterministic irreducible polynomial generation?

In $\Bbb F_q[x]$ given $d\in\Bbb N$ there is a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=deg(x)$ under $GRH$ and an unconditional randomized algorithm. Do ...
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0answers
72 views

Scaled down and scaled up versions of Impagliazzo-Wigderson Therem

A famous theorem due to Impagliazzo and Wigderson states that if some function in $E=DTIME[2^{O(n)}]$ requires circuits of size $2^{\Omega(n)}$ then P=BPP. When can we change $P$ with some ...
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43 views

What is the best known gap between ZPP and Deterministic communication complexity? [duplicate]

I know that $N(f) \times coN(f) \geq D(f)$. This means that $ZPP(f) \geq \sqrt{D(f)}$. Is this separation tight?
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1answer
185 views

What do stronger circuit lower bounds give in terms of derandomization?

We have $EXP\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\epsilon})$ at every $\epsilon>0$. This is essentially $DTIME(2^{O(n)})\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\...
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155 views

Why are one way functions and pseudorandom number generators considered necessary or essential for derandomization?

If strong pseudorandom number generator exists then $BPP=P$ holds and if one way functions exists then $BPP\subseteq SUBEXP$ holds. What are the best statements we have proved that come close to ...
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618 views

Is BPP= P known for ANY uniform model of computation?

Many believe that BPP $=$ P "should" hold for Turing machines. We even have some "witnesses" for this: otherwise some "strange" things would happen; see e.g. this paper by Implagliazzo and Wigderson. ...
12
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1answer
471 views

Is BPP vs. P a real problem after we know BPP lies in P/poly?

We know (for now about 40 years, thank Adleman, Bennet and Gill) that the inclusion BPP $\subseteq$ P/poly, and an even stronger BPP/poly $\subseteq$ P/poly hold. The "/poly" means that we work non-...
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365 views

Fine-grained complexity of BPP

If E does not have i.o.-$2^{o(n)}$ circuits, then P=BPP, but this does not tell us about the fine-grained containments between $\mathrm{Time}(n^a)$ and $\mathrm{BPTime}(n^b)$. Are there reasonable ...
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106 views

On approximating problems in $\#P$

We know that for every counting problem $\#A$ in $\#P$, there is a probabilistic algorithm $\mathcal C$ that on input $x$, computes with high probability a value $v$ such that $$(1 − ε)\#A(x) ≤ v ≤ (1 ...
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1answer
115 views

On $\Delta_i^P$

We know $P\subseteq NP\cap coNP\subseteq\Delta_i^P=P^{\Sigma_{i-1}^P}\subseteq \Sigma_i^P\cap\Pi_i^P=NP^{\Sigma_{i-1}^P}\cap coNP^{\Sigma_{i-1}^P}$. If $P=BPP$ is there a 'higher' randomized class ...
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155 views

On earlier references for $P=BPP$ and Kolmogorov's possible view on modern breakthroughs involving randomness?

Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate P=BPP in 1986. They do this without getting into circuit lower bounds and from a different view which ...
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98 views

Deterministic approximation algorithms for treewidth

As far as I understand, the factor $O(\sqrt{\log OPT})$ approximation algorithm for treewidth of Feige, Hajiaghayi, and Lee is randomized, and no deterministic approximation algorithm with this factor ...
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0answers
89 views

P=BPP and derandomizing Vazirani-Valiant?

Vazirani-Valiant reduction is a randomized reduction from SAT to unambiguous SAT. Is P=BPP strong enough to derandomize Vazirani-Valiant reduction? If not what other ingredients are necessary to ...
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1answer
120 views

Unambiguous SAT and sparse languages

What is the consequence if there are only polynomially many 'yes' classes of instances of a language that is polynomial time reducible from a problem equivalent to UnambiguousSAT (such as possibly ...
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1answer
412 views

Implications of faster randomized $CIRCUIT SAT$ algorithm

In here on page $13$ proposition $1$ it says 'If $CIRCUIT$ $SAT$ on $n$ inputs and $m$ gates is in $2^{n^{o(1)}}poly(m)$ time, then $EXP\not\subseteq P/poly$'. Can we have randomized $2^{n^{o(1)}}...
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1answer
172 views

Uniform derandomisation of circuit complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined in the same way as $\textrm{BPP}$ is defined with respect to $\textrm{P}$. ...
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1answer
289 views

Adleman's theorem over infinite semirings?

Adleman has shown in 1978 that $\mathrm{BPP}\subseteq \mathrm{P/poly}$: if a boolean function $f$ of $n$ variables can be computed by a probabilistic boolean circuit of size $M$, then $f$ can be also ...
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226 views

On the shortest vector problem (is it $NP$-complete?)

Ajtai has shown that shortest vector problem is $NP$-hard by using randomized reduction from subset sum. Has this been derandomized?
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1answer
107 views

Notion similar to k-wise independence

I want to construct a family of functions $H:\{0,1\}^n \rightarrow \{0,1\}$ with a property that is similar to k-wise independence. Specifically, I want $H$ to satisfy the following property. Let $k$ ...
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2answers
647 views

Is it known whether $BPP\cap NP\subseteq RP$?

The reverse inclusion is obvious, as is the fact that any self-reducible NP language in BPP is also in RP. Is this also known to hold for non-self-reducible NP languages?
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165 views

Derandomization of Polynomial Identity Testing

There are some theorems that state $P = BPP$ if some condition is satisfied. For example, a theorem of Impagliazzo and Wigderson states tha $P=BPP$ unless $DTIME(2^{O(n)})$ has sub-exponential ...
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3answers
902 views

Examples of successful derandomization from BPP to P

What are some major examples of successful derandomization or at least progress in showing concrete evidence towards $P=BPP$ goal (not the hardness randomness connection)? The only example that comes ...
2
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1answer
140 views

Minimum weights needed to derandomize weight assignment by isolation lemma

Under isolation lemma if you have a graph with $2n$ vertices and $m$ edges an isolating weight assignment can be obtained by assigning edges weights randomly from $\{1,2,\dots,2m-1,2m\}$. A weight ...
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1answer
185 views

Connections between Graph Isomorphism and Polynomial Equivalence

Are there any relations between Graph Isomorphism problem and Polynomial Equivalence problem? In particular does a polynomial time solution to Graph Isomorphism problem provide any evidence towards ...
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1answer
247 views

Smallest $f(n)$ such that $P/f(n) = BPP/f(n)$?

It is well-known that $\mathsf{P/poly}(n) = \mathsf{BPP/poly}(n)$. It is a major open problem to prove the conjecture $\mathsf{P} = \mathsf{BPP}$. $\mathsf{P} = \mathsf{BPP}$ implies $\mathsf{P}/f(n) ...
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136 views

Consequences of VP = VNP on randomness

According to the answers in posting it is possible that $\mathsf{VP} = \mathsf{VNP}$ and $\mathsf{P} \neq \mathsf{NP}$ are simultaneously correct. $\mathsf{VP} = \mathsf{VNP}$ implies $\mathsf{P/...
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What is worst case complexity of number field sieve?

Given composite $N\in\Bbb N$ general number field sieve is best known factorization algorithm for integer factorization of $N$. It is a randomized algorithm and we get an expected complexity of $O\Big(...
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2answers
299 views

Consequence of PIT over $\Bbb Z[x_1,\dots,x_n]$ not having efficient algorithm

Given $p(x_1,\dots,x_n),q(x_1,\dots,x_n)\in \Bbb Z[x_1,\dots,x_n]$ such that coefficients of $p,q$ are bounded by $B$, does $p\equiv q$ hold? Schwartz-Zippel lemma applies here since it holds for ...
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1answer
325 views

What is the status of intermediate problems if P is not NP in worst way imaginable?

Assume $P\neq BPP\neq NP$ with caveat that there is a deterministic algorithm for every $NP$ complete problem with input size $n$ bits in $2^{(\log n)^{1+f(n)}}$ arithmetic operations on $\log n$ ...
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1answer
223 views

Is there a randomized complexity class analogous to $\mathsf{P/Poly}$?

$\mathsf{P/Poly}$ captures those problems that could be solved in polynomial time given some precomputed polynomial number of constants. Is there an analogous complexity class in randomized world ...
10
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1answer
255 views

Can we construct a k-wise independent permutation on [n] using only constant time and space?

Let $k>0$ be a fixed constant. Given an integer $n$, we want to construct a permutation $\sigma \in S_n$ such that: The construction uses constant time and space (i.e. preprocessing takes ...
12
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1answer
218 views

When does randomization stops helping within PSPACE

It is known that adding bounded-error randomization to PSPACE doesn't add power. That is, BPPSAPCE=PSPACE. It is famously unknown whether P=BPP, but it is known that $BPP\subseteq \Sigma_2\cap \Pi_2$....
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1answer
249 views

On derandomizing polynomial identity testing

In polynomial identity testing we seek a deterministic algorithm to infer equality of two polynomials $g,h\in\Bbb Z[x_1,\dots,x_n]$. Derandomizing known efficient randomized algorithms and producing ...
2
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1answer
199 views

Extractors in Practice: How to Determine the Min-Entropy in the Source Distribution

One of the main parameters in the construction of extractors is $k$, the min-entropy of the source distribution. In practice, suppose we want to extract randomness from a given source $S$. How do we ...
5
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1answer
131 views

Why does the transformation in the proof for SL=L preserve connectedness of s and t?

I'm currently working on this paper showing that $\mathcal{SL} = \mathcal{L}$. But I keep wondering how the transformation $\tau$ in definition 3.1 preserves the connectivity of $s$ and $t$. $\tau$ ...
27
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3answers
822 views

Problems in $\mathsf{BPP}$ not known to be in $\mathsf P$?

What problems are known to belong to $\mathsf{BPP}$ but not known to belong to $\mathsf P$? More precisely, I am interested in independent problems, that is whose derandomizations are not known to be ...
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1answer
110 views

Hitting set of very restricted linear forms

We say that $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$ is a {-1,0,1}-linear form if $f=\sum_{i\in S}x_{i}-\sum_{i\in T}x_{i}$ where $S,T\subseteq[n]$. A hitting set $H\subseteq\mathbb{Z}^{n}$ for {-1,0,1}-...
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268 views

Extensions of Affine Dispersers

A function $f\colon\{0,1\}^n\to\{0,1\}$ is called an affine disperser for dimension $d$, if for every affine subspace $S\subseteq \{0,1\}^n$ of dimension at least $d$, $f$ is not constant on $S$. This ...
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164 views

Randomized Parallel Algorithm for Maximal Independent Set

There are a couple of randomized parallel algorithms for the maximal independent set problem, e.g. A Simple Parallel Algorithm for the Maximal Independent Set Problem, A fast and simple randomized ...
4
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1answer
191 views

Arithmetic Analogues of P versus BPP

In the arithmetic hierarchy, is there an analog of $P$ versus $BPP$? Particularly is there a notion of randomness there? If there is no such analogy, why is randomness in the resource bounded case ...
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217 views

Physical Proof for P versus BPP

Lipton asks for a physical proof of $P\neq NP$. Can we even ask for a physical proof for understanding $P=BPP$ or $P\neq BPP$? Is there anything in physics that lets us avoid randomness? ...
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224 views

How to derandomize the Chernoff bound?

Avi Wigderson have a paper on how to derandomize the matrix-valued Chernoff bound. I would like to know whether there exists a simple version of paper on how to derandomize the real-valued Chernoff ...
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1answer
749 views

Impagliazzo and Wigderson's famous P=BPP paper

I'm reading Impagliazzo and Wigderson's famous $\mathsf P=\mathsf{BPP}$ paper in 1997. Since I'm new to this field and the paper is a concise conference version, I have difficulty following their ...
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133 views

Concentration Bounds for Dependent Rounding

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...