Questions tagged [complexity-classes]

Questions about relationships between complexity classes.

Filter by
Sorted by
Tagged with
3
votes
0answers
25 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
votes
0answers
18 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
votes
0answers
26 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
37 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
66 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
62 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
31 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
30 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
60 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 ...
4
votes
0answers
92 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
94 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
58 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
69 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
19 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
52 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
17 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
39 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
23 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
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 ...
1
vote
2answers
75 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
125 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
28 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
47 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
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 ...
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
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\...
3
votes
1answer
79 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 $PSPACE\subseteq DTIME[n^{O(\log n)}]$?
0
votes
0answers
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 ...
1
vote
1answer
32 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
49 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
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 ...
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
48 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^...
0
votes
0answers
35 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-...
0
votes
1answer
24 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 $NC^1=L$?
2
votes
1answer
58 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
votes
1answer
227 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: $a^b$
0
votes
1answer
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 ...
0
votes
1answer
217 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 ...
1
vote
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 :)
0
votes
1answer
21 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.
1
vote
1answer
111 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 ...