Questions tagged [complexity-classes]
Questions about relationships between complexity classes.
341
questions
3
votes
1answer
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 ...
0
votes
0answers
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 ...
2
votes
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), ...
0
votes
1answer
28 views
how to proof ${ NPC \bigcap CO-NPC \ne \varnothing then NP = P ? }$
how proof ${\ \ NPC \ \ \bigcap \ \ CO-NPC \ne \varnothing }$ then
${NP = P ? }$
2
votes
1answer
44 views
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 ...
1
vote
0answers
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?
5
votes
2answers
76 views
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
votes
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 ...
0
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0answers
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-...
3
votes
0answers
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 ...
0
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0answers
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 ...
0
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0answers
32 views
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 ...
0
votes
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 ...
1
vote
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 $p(n)$?
2
votes
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?
$$ L = \{0,1\}^∗ ∈ P? $$
1
vote
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....
1
vote
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 ...
0
votes
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 ...
5
votes
0answers
97 views
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
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
0answers
21 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 ...
1
vote
0answers
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 ...
0
votes
1answer
76 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 ...
1
vote
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 ...
0
votes
0answers
18 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 $k\geq0$?
0
votes
0answers
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?
0
votes
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 $L ...
2
votes
0answers
45 views
Is PSPACE vs NEXPTIME known?
I know that P = PSPACE is a famous open problem, and that EXPTIME = NEXPTIME is also unknown. By the time heirarchy theorem we know that NP is a strict subset of NEXPTIME.
Is anything known about ...
1
vote
1answer
24 views
Problem class of assigning N persons to N tasks, zero costs with prefs
I am looking for the general problem class / computational complexity / algorithms for the following problem:
N tasks must be accomplished by N persons. 1 task to be done by exactly 1 person and vice ...
1
vote
2answers
20 views
Why do we use worse-case when categorising problems?
Maybe I am wrong, but I read that when we categorise problems in their respective complexity classes, we use worse-case analysis.
Why don't we use the average case?
I imagine we could have a problem ...
1
vote
2answers
78 views
DSPACE(f(n)) closed under complement
I think you can create the complementary language that is an element of DSPACE($f(n)$), where $f(n) \geq \log(n)$ by adding a step to the algorithm that reverses the answer. By that the function $f(n)$...
0
votes
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
votes
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\ ...
-1
votes
1answer
52 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
2
votes
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}'\...
1
vote
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\...
5
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
1
vote
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 $f(n)$, ...
1
vote
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 ...
1
vote
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?
1
vote
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 ...
1
vote
0answers
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 $P$...
0
votes
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^...
