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Questions tagged [derandomization]

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

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3
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2answers
144 views

If $P=BPP$, then Is it correct that $IP=NP$?

This is my first question in this site. I ask this question since I got no comment and no answer for one year and two months in cs.stackexchange and it was automatically deleted by the system. So, ...
1
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1answer
85 views

Efficient randomness reduction using k-wise independence

Consider a randomized algorithm with runtime $n$, which succeeds with high probability. The algorithm uses $O(n)$ uniformly random bits. Now it is given that we can replace these uniformly random ...
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0answers
170 views

P=BPP and derandomizing Vazirani-Valiant?

Vazirani-Valiant reduction is a randomized reduction from $SAT$ to unambiguous $SAT$. 1. Is $P=BPP$ strong enough to derandomize Vazirani-Valiant reduction? 2. If not what other ingredients are ...
2
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1answer
136 views

Optimal bounds for $k$-wise non-uniform random bits

Let $k\geq 2$ be a constant (in my case, $k=4$), and $n,t \geq 0$ be integers such that $2^t \leq n$. What is the smallest sample space (or, equivalent, how many true independent random bits are ...
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0answers
247 views

On $BPP$ in $P^{NP}$ and $SETH$

It is believed showing $BPP$ in $P$ involves good $PRG$s and faces lower bound barriers. Does showing $BPP$ in $P^{NP}$ which would mean $BPP\neq EXP^{NP}$ face similar $PRG$ and give lower bounds? ...
2
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0answers
53 views

How to improve this pseudorandom generator?

Let $f$ be a Boolean function and $\varepsilon > 0$. There exists a pseudorandom generator $G_f: \{0,1 \}^{n^{\varepsilon}} \to \{0,1 \}^n$ with the following property. Let $T$ be a set and $...
6
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1answer
323 views

Error reduction with expanders and derandomization

In "standard" error reduction with an expander, if a randomize algorithm uses $n^d$ random bits, we need $n^d+O(n)$ random bits to achieve $2^{-O(n)}$ error probability. Now, if the algorithm has a ...
3
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1answer
148 views

Algebraic construction of $\varepsilon$-biased sets

Let $\ell> 1$ be an integer and consider the mapping $\text{Tr}:\mathbb{F}_{2^\ell}\to\mathbb{F}_{2^\ell}$ defined by $$\text{Tr}(x)=x^{2^0}+x^{2^{1}}+\cdots+x^{2^{\ell-1}}$$ It is then possible to ...
3
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1answer
149 views

Lower bound on the support size of an $\epsilon$-biased distribution

Let $D$ be an $\epsilon$-biased distribution we want to show that $$\text{Supp}(D)\geq \Omega\bigg(\frac{n}{\epsilon^2\log(\frac{1}{\epsilon})}\bigg)$$ I know that there are some proofs for this but I ...
0
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1answer
119 views

Permuting the columns of a 0/1-matrix to avoid short segments

Consider an $n \times n$ table with $n$ stars such that each row contains at most $\log n$ stars. The stars break each row into segments (continuous parts of a row without stars). Let's call a segment ...
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0answers
56 views

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\}^...
5
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1answer
243 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 ...
8
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1answer
98 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 ...
3
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0answers
100 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 ...
12
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2answers
2k views

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(...
5
votes
1answer
233 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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0answers
56 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 ...
2
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0answers
75 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 ...
5
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1answer
268 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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0answers
45 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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183 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 ...
0
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1answer
190 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^\...
24
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2answers
827 views

Space efficient “industrial” unbalanced expanders

I am looking for unbalanced expanders that are "good" and "space-efficient". Specifically, a bipartite left-regular graph $G=(A,B,E)$, $|A|=n$, $|B|=m$, with left degree $d$ is a $(k,\epsilon)$-...
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0answers
394 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 ...
29
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0answers
715 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. ...
2
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0answers
107 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 ...
16
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1answer
960 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-...
2
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1answer
125 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 ...
4
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0answers
170 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 ...
6
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0answers
104 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 ...
1
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1answer
141 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 ...
5
votes
1answer
420 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)}}...
9
votes
1answer
200 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 $\textrm{P}$. ...
13
votes
1answer
314 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 ...
6
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0answers
307 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?
3
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1answer
112 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 $k$ ...
15
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3answers
1k 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
145 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 ...
10
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2answers
771 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?
2
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0answers
171 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 ...
11
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2answers
307 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 ...
63
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1answer
3k views

More on PH in PP?

A recent question by Huck Bennett asking whether the class PH was contained in the class PP, received somewhat contradictory answers (all true, it seems). On one hand, several oracle results were ...
4
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1answer
196 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 ...
0
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0answers
144 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/...
25
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4answers
1k views

What specific evidence is there for P = RP?

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by ...
-2
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1answer
329 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 $\log n$ ...
-2
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1answer
229 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
283 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
225 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 $BPP\subseteq \Sigma_2\cap \Pi_2$....
10
votes
1answer
287 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 ...