Questions tagged [complexity-classes]

Questions about relationships between complexity classes.

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Is DISCRETE LOG a NP hard problem?

In cryptography there are two problems which are part of the foundation of modern public key cryptography. Both of them can be solved in polynomial time on quantum computers. I am talking about: FACT ...
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34 views

If P = NP then EXPTIME = NEXPTIME?

Are the following implications correct? If P = NP then EXPTIME = NEXPTIME If P $\neq$ NP then NEXPTIME = 2EXPTIME If so would the corresponding implications hold for N2EXPTIME, etc.?
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Big-O Notation and Calculus?

I was wondering if there are any calculus relationships implicit in Big-O notation. For example, an algorithm linear according to Big-O notation reduces the size of the problem by a constant amount ...
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1answer
63 views

Does EXP^EXP = EXP? [duplicate]

Does $\mathrm{EXP}^\mathrm{EXP}=\mathrm{EXP}$? Here is my thought: $\mathrm{EXP}$ machine can ask $2^{O(n)}$ queries to the oracle, and each oracle would itself solve an exponential time problem in a ...
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17 views

Which is harder, an NP-complete problem or the Raz-Tal oracle problem?

This is a (hopefully) sharper version of a question that I asked previously. Which of these algorithms is believed to have a longer asymptotic runtime? The optimal algorithm guaranteed to solve some ...
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14 views

Do relativized relations between complexity classes tell us anything about the nonrelativized relation?

The existence of relativized relations between complexity classes seems to often be treated as "circumstantial" evidence about the "true" or "real-world" (i.e. nonrelativized) relation between the ...
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1answer
46 views

Proving that $NPSPACE\subseteq PSPACE$ using the proof of Savitch's Theorem

We were shown a proof of $NPSPACE\subseteq PSPACE$ in class. In short, the proof says: Let $L\in NPSPACE$. Then there exists a non-deterministic polynomial space bounded Turing machine $M$ that ...
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1answer
35 views

Complexity of K-Colorful Coloring Problem for a Hypergraph

I searched a lot trying to find a reference for the complexity of K-colorful coloring problem for a hypergraph but I cannot find it. Please if anyone has a reference for the complexity of the problem ...
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14 views

$P$ with $SAT[k]$ and $NP[k]$ oracles?

We know $coNP$ is in $P^{NP}$ and so does $coNP$ in $P^{NP[1]}$ and $P^{SAT[1]}$ hold? Is there a difference between $P^{SAT[k]}$ and $P^{NP[k]}$ at any $k\geq0$?
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43 views

Intersection of decision problems?

Say we have two problems $\Pi_1\in NP$ and $\Pi_2\in coNP$. Where does $\Pi_1\cap\Pi_2$ live?
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43 views

prove that there is a complete language in $L \cup \{A_{TM}\}$

$A_{TM} = \{\langle M,w\rangle\mid w\in L(M)\}$ $L$ = complexity class containing decision problems that can be solved by a deterministic Turing machine using logarithmic space Given the language $L ...
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33 views

Is PSPACE vs NEXPTIME known?

I know that P = PSPACE is a famous open problem, and that EXPTIME = NEXPTIME is also unknown. By the time heirarchy theorem we know that NP is a strict subset of NEXPTIME. Is anything known about ...
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1answer
22 views

Problem class of assigning N persons to N tasks, zero costs with prefs

I am looking for the general problem class / computational complexity / algorithms for the following problem: N tasks must be accomplished by N persons. 1 task to be done by exactly 1 person and vice ...
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2answers
19 views

Why do we use worse-case when categorising problems?

Maybe I am wrong, but I read that when we categorise problems in their respective complexity classes, we use worse-case analysis. Why don't we use the average case? I imagine we could have a problem ...
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2answers
72 views

DSPACE(f(n)) closed under complement

I think you can create the complementary language that is an element of DSPACE($f(n)$), where $f(n) \geq \log(n)$ by adding a step to the algorithm that reverses the answer. By that the function $f(n)$...
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1answer
122 views

Proving NP completeness of maximal length path

I have this question to answer: For each node i in an undirected network $G = (N,E)$, let $N(i) = \{j \in N : \{i, j\} \in E\}$ denote the set of neighbors of node $i$ and let $c_e\geq0$ denote the ...
2
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1answer
26 views

NL-Hardness of Target

When revising for an upcoming exam in complexity theory, I came across this problem on the final part of a question, which I was unable to solve: $ TARGET = \{<G, t> : t\ is\ reachable\ from\ ...
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34 views

A simple clarification on polynomial hierarchy

$P^{NP}\subseteq BPP^{NP}$ holds. According to current knowledge $BPP$ is in $\Sigma_2^P\cap\Pi_2^P$ holds. So according to current knowledge is following true? $P^{\Sigma_2^P\cup\Pi_2^P}\subseteq ...
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18 views

Why is $BPP^{NP}$ in the polynomial hierarchy?

Why is $BPP^{NP}$ in the polynomial hierarchy? I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. Should I just simply reuse the proof of $BPP\subset NP^{NP}$, feed in each machine ...
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30 views

is this given language in class P? [duplicate]

i was wondering: is this language in class P? $NONDISJOINT_{DFA}\:=\:\left\{<A,B> |\:A\:and\:B\:are\:DFAS\:and\:L\left(A\right)\:\cap L\left(b\right)\:\ne \varnothing \right\}$ explanation: a ...
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1answer
27 views

Can CNF with an input string be evaluated in logarithmic space?

I have been trying to solve satisfiability of {$<c, w>$ | $c$ is a CNF and $w$ is a binary string which satifies the $c$}. As first looks to me, it is satisiable in linear time ($O(n)$) since ...
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1answer
35 views

On the robustness of BQP class

Typically the notion of quantum Turing machine is introduced with its transition function. $$ \delta:Q\times \Gamma\rightarrow \mathbb{C'}^{Q\times \Gamma\times\{L,R,0\}} $$ Where $\mathbb{C}'\...
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1answer
24 views

Changing probabilities to 0/1 in definition of class IP

A language $L$ belongs to $\mathbf{IP}$ if there exists $V,P$ such that for all $Q$, $w$, $$w\in L\Rightarrow Pr[V\leftrightarrow P\text{ accepts }w]\geq2/3$$ $$w\notin L\Rightarrow Pr[V\...
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1answer
70 views

If $NP\subseteq DTIME[n^{O(\log n)}]$ then what happens?

If $NP\subseteq DTIME[n^{O(\log n)}]$ then what happens? Does it imply $NP\neq EXP$? Is there any other consequences such as $BPP\neq EXP$? Does it also give $PSPACE\subseteq DTIME[n^{O(\log n)}]$?
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37 views

Reduction of complement from complexity class co-np and p

Let P $ \neq $ NP. D is in the complexity class co-NP. B is in the complexity class P. Let $ \bar{D} $ be the complement of D, then $\bar{D} $ $\leq _ {p} $ B. Is this statement true or false? My ...
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1answer
29 views

Show: “Checking no solution for system of linear equations with integer variables and coefficients” $\in \mathbf{NP}$

I've been struggling for a while trying to solve this problem: Show that the following problem is in $\mathbf{NP}$: Check that a system of linear equations with $m$ integer variables and integer ...
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1answer
43 views

Proof of proposition between Precise Turing Machine and Proper Complexity function

In "Computational Complexity" textbook by C. H. Papadimitriou, p. 141, he proved the following claim. Proposition 7.1: Let there be a DTM/NDTM M that decides a language L within time/space $f(n)$, ...
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2answers
35 views

NP-hardness does not imply lower bound, strictly speaking?

A problem is NP-hard iff every NP problem can be polynomially-time reduced to it. Hardness is often intuitively explained as a lower bound. But it isn't, strictly ...
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8 views

$NC$ and $FNC$ oracles low for functional and Stockemeyer classes respectively?

We know $P^{NC}=P$ and $FP^{FNC}=FP$ hold. Do $FP^{NC}=FP$ and $P^{FNC}=P$ hold?
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1answer
11 views

Alternative formulation of complexity class $BPP$

In Aurora and Barak, they give the following alternative definition of $BPP$: What is the meaning of the subscript to $Pr$? Is that $Pr_{r \in_R \{0,1\}^{p(|x|)}}$? My guess is this is supposed to ...
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16 views

Other problems in UP and co-UP

Are there any known problems in $UP \cap co-UP$ other than integer factorization and parity games (or a problem that can be reduced in polynomial time to either problem), that aren't known to be in $P$...
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1answer
47 views

Why all problems are not in P complexity?

İf Somebody wants to play perfect chess.I consider for finding solution to this problem,possible minimum data compression method which describes with 10 bit maximum possible data is in the range of 2^...
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21 views

Complexity class without fixed-poly size circuit

$PP$ is shown to have no fixed-poly size circuit by Vinodchandran. Bounded inside the polynomial hierarchy, $\Sigma^2_p$ is also shown to possess no fixed-poly size circuit by Kannan. In notation, ...
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32 views

$\mathbb{NEXP\subseteq(NEXP\cap coNEXP)/poly}\implies \mathbb{NEXP=NEXP\cap coNEXP}$

We already know that $NEXP\subseteq EXP/poly\implies NEXP=EXP$. What if we change $EXP$ to $NEXP\cap coNEXP$. As the title states, prove the statement in the title. The original proof use the self-...
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1answer
23 views

What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?

What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?
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1answer
48 views

Relationship between complexity classes XP and W[1]?

I am reading the introductory chapter in Parameterized Algorithms by Cygan et al. and I am having some problems with the distinction between complexity classes $\mathsf{W[1]}$ and $\mathsf{XP}$. ...
4
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1answer
225 views

Is exponentiation in P?

I think the following problem belong to class P, but I don't know how I can prove it, could somebody help me? Inputs: two numbers $(a,b) \in \mathbb{N}$ Output: $a^b$
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36 views

Will such an algorithm be feasible, will it be considered a poly time algorithm?

Given a p-digit number n, where p is atleast 500, if we have an algorithm whose computational complexity is in the worst case, $O(p\mathrm{log}(p))$, will it be counted as a poly time algorithm ? Or ...
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1answer
168 views

Modular exponentiation running time

I read on Wikipedia that modular exponentiation can be done in polynomial time. I've a few questions regarding it (sorry if they seem a bit easy – I'm not a comp sci student). Is it poly ...
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1answer
34 views

If a language is contained in other langauge, is it of the same complexity?

If some language $L$ is in P, and some other language $K$ is contained in $L$, does that mean that $K$ is also in P? Thanks :)
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1answer
19 views

$\mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{P}/\mathrm{poly} \implies \mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{MA}$

Is anyone able to give a concise proof for the implication stated in the title? This is gonna be in stark contrast to this question. For definition of $\mathrm{strict}$-$\mathrm{SUBEXP}$, see here.
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1answer
108 views

P=NP giving a deterministic algorithm for SAT

I'm trying to prove the following problem: Prove that if $P=NP$ then there is a polynomial time algorithm for the following problem: INPUT: A boolean formula $\phi$. OUTPUT: A satisfying ...
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1answer
25 views

Hardness of $2$ edge-disjoint spanning trees decomposition

The question is clear from the title. What is the complexity of the following decision problem: Input: An undirected graph $G(V, E)$ Output: $\mathrm{YES}$ if $G$ can be decomposed into two ...
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1answer
110 views

Show that RP is closed under concatenation

I'm trying to prove the following problem: Show that $RP$ is closed under concatenation Now, let's say that the two languages are $L_{1}$ and $L_{2}$ (both in $RP$). Then I accept a word iff the ...
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1answer
30 views

Complexity classes that are low for equivalent definitions of $\mathrm{PP}$

What is the biggest complexity class that is low for each other equivalent definition of $\mathrm{PP}$? I already know that $\mathrm{PP}^\mathrm{BQP}=\mathrm{PP}$. This is a lowness result using ...
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1answer
59 views

Looking for a problem provably not in P

My basic position is that everything is in P. Then comes the time hierachy theorem and EXP. That's easy: simulate and then diagonalize. After that comes EXP-completeness; that's difficult to swallow. ...
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1answer
28 views

Is there a complexity class “BQP without error”?

I was wondering if there is a complexity class for problems that can be solved efficiently by a quantum computer such that it always gives the right answer? For example the Deutsch-Josza algorithm ...
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26 views

What if $PSPACE$ falls to non-uniformity?

We know if every language in $EXP$ has polysize circuits, then $P\neq NP$ and $EXP=PSPACE=\Sigma_2^P\cap\Pi_2^P$. If every language in $PSPACE$ has polysize circuits, then does it give anything (note ...
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1answer
103 views

Is $\mathsf{DSPACE}(n)=\mathsf{DSPACE}(n/\log\log n)$?

We know that $\mathsf{DSPACE}(\log\log n) = \mathsf{DSPACE}(1)$ according to this proof. Can we claim that $\mathsf{DSPACE}(n)=\mathsf{DSPACE}(n/\log\log n)$ or something like $\mathsf{DSPACE}(n^3)=\...
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1answer
40 views

Question on time hierarchy [closed]

How can I prove that $\mathsf{DTIME}^{\mathrm{Htm}}(n^2)$ is contained in $\mathsf{DTIME}^{\mathrm{Htm}}(n^3)$? (sorry about how it is written. I mean the set of languages decided by a deterministic ...