Questions tagged [complexity-classes]

Questions about relationships between complexity classes.

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50 views

Comparing asymptotic running time of two algorithms $\sqrt n$ and $2^{\sqrt{\log _{2}n}}$

Given two algorithms with their time-complexity $t_a(n)=\sqrt{n}$ and $t_b(n) = 2^{\sqrt{\log _{2}n}}$ and i have to show $t_b(n) = O(t_a(n)) $. I´ve made a program to check this statement and it ...
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20 views

Proof by padding: $\textsf{TIME}(t_1(n)) = \textsf{TIME}(t_2(n)) \implies \textsf{TIME}(t_1(f(n))) = \textsf{TIME}(t_2(f(n)))$

I've been given the task of proving the statement in the title, which I found out it should be called the translational lemma by means of a padding argument; $f$, $t_1$ and $t_2$ are three ...
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0answers
54 views

Proving a pattern exist in a string without revealing where

Some time ago i read the following problem (i don't remember the article from which i read it from) : "Suppose you are given a picture where the goal is to find waldo (from the game where is waldo), ...
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1answer
28 views

how to proof ${ NPC \bigcap CO-NPC \ne \varnothing then NP = P ? }$

how proof ${\ \ NPC \ \ \bigcap \ \ CO-NPC \ne \varnothing }$ then ${NP = P ? }$
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1answer
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Assume that NP = DTIME(2^sqrt(n)), prove that DTIME(2^sqrt(n)) = DTIME(2^n)

I tried using the padding argument to prove such a thing (as it appeared in Arora's book), but I am not sure how this technique will help me here. I am trying to get to a contradiction to the Time ...
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32 views

Complexity problem reduction?

Let say A and B are two decesion problems where A $\le$ B polinomial reduction is true. Is this : A̅ $\le$ B̅ also true? If so, can you show an exemple, if not why?
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Does $P/O(1)$ equal to $P$ if solver needs to consider smaller inputs?

Suppose that $F$ is a problem such that for every $n$, there is a program of length $O(1)$, running in polynomial time to $n$, that solves $F$ correctly on all instances of size less than $n$. Can $F$ ...
3
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1answer
99 views

Grover's algorithm on probabilistic classical machines

As a starting point for this question, I came across this question, which AIUI is citing a construction showing how to simulate quantum circuits with a $PP$ algorithm, i.e. implying quantum ...
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30 views

A heuristic for finding an edge cycle cover

I am looking to find a minimum list of cycles in a graph such that their union gives the list of all simple cycles in this graph. In the example below, here are 4 simple undirected cycles: 1-2-3, 2-3-...
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32 views

Approximate algorithms for class P problems

As a part of my Algorithm course we studied Approximate Algorithms for NP-complete or NP-hard problems, e.g. "set cover", "vertex cover", "load balancing", etc. My professor asked us as an extra ...
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27 views

Cook Levin Theorem (Sipser Proof) (phi move)

In Sipser's proof of the cook levin Theorem the move function (phi move) checks whether a given window is legal. For that we must have an exhaustive set of all possible legal windows to verify that a ...
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Why is $DSPACE(\log(n)) = NSPACE(\log(n))$ not known?

Here $DSPACE(\log(n))$ is the family of algorithms for which there exists a deterministic Turing machine using $O(\log(n))$ space. On the other hand $NSPACE(\log(n))$ is the family of algorithms for ...
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1answer
45 views

If a sparse language is NP complete, then are all languages in NP in P/poly?

If a Sparse Language is NP complete, then are all languages in NP in P/poly? I know that sparse languages are in P/poly, but does a polynomial time reduction give an addition to the circuit that is ...
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1answer
69 views

If a sparse language is NP-complete then are all NP languages sparse?

If a Sparse Language is NP-complete then are all NP languages sparse? We say a language is sparse if $\forall n \in \mathbb{N}, |L \cap \{ 0,1 \} ^{n}| \leq p(n)$, for some polynomial $p(n)$?
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1answer
67 views

{0,1}* ∈ P class?

I have the following question about complexity time classes. Given the language $L = \{0,1\}^*$, is it inside the class P or not? $$ L = \{0,1\}^∗ ∈ P? $$
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1answer
48 views

If an NP complete problem 'A' is polynomial time reducible to another problem 'B' does that imply 'B' is also NP complete?

The following question was asked on a quiz: Let S be an NP-complete problem, and Q and R be two other problems (that we don't know much about). If we now know that Q is polynomial time reducible (i....
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1answer
52 views

Why log-space reduction is used for NL-completeness while PSPACE reduction isn't used for PSPACE completeness?

NL-Complete languages are defined by Log-space reduction, while PSPACE complete languages are defined by poly-time many-to-one reduction. According to these posts : Why not polynomial-space ...
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2answers
118 views

Reducing from NPC to Co-NPC => NP = Co-NP?

In my lecture we learned: If X is NPC and X in Co-NP => NP = Co-NP Would it be enough to prove NP = Co-NP if I reduce a ...
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Is there a polynomial-time algorithm to minimize regular expressions without Kleene closures/stars?

I have read that minimizing regular expressions is, in general, PSPACE-complete. Is it known whether minimizing regular expressions without the Kleene closure (star, asterisk) is in P? The language ...
4
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1answer
240 views

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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2answers
103 views

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
76 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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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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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
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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
36 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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$P$ 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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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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1answer
44 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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45 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
24 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
20 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
78 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
127 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
31 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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1answer
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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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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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31 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
31 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
25 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
106 views

If NP is a subset of DTIME[n^O(log n)] then what happens?

If $\mathsf{NP}\subseteq \mathsf{DTIME}[n^{O(\log n)}]$ then what happens? Does it imply $\mathsf{NP}\neq \mathsf{EXP}$? Is there any other consequences such as $\mathsf{BPP}\neq \mathsf{EXP}$? Does ...
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40 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
40 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
60 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
38 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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0answers
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
12 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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0answers
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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
49 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^...