Questions tagged [complexity-classes]

Computational complexity classes and their relations

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5
votes
2answers
595 views

XOR-SAT to Horn-SAT reduction

Horn-SAT (conjunction of Horn clauses) is P-complete and XOR-SAT (conjunction of xor clauses) is in P. This means that there is a reduction from XOR-SAT to Horn-SAT weaker than a polynomial reduction. ...
11
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1answer
294 views

$\mathsf{NP^{PP}}$ vs $\mathsf{P^{PP}}$

Is $\mathsf{NP^{PP}} = \mathsf{P^{PP}}$? Or, more generally, Is $\mathsf{NP^{PP}} \subseteq \mathsf{P^{PP}/poly}$?
5
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0answers
134 views

An oracle in $\mathsf{NEXP}$ that separates ZPP from BPP

Does there exist an oracle $A \in \mathsf{NEXP}$ such that $ \mathsf{ZPP}^A \neq \mathsf{BPP}^A$?
2
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1answer
248 views

Complexity involving connected components of 0/1 matrix

Assume a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where each step you take you ...
4
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0answers
178 views

Subexponential algorithms vs separations

Williams (see also slide 29 here) has shown that an $\frac{2^{n}}{n^{10}}$ algorithm for satisfiability of circuits belonging to a class $\mathcal{C}$ imply that $\mathrm{NEXP}\nsubseteq \mathcal{C}$. ...
7
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1answer
155 views

What are the hardness results known for CSP over $\mathbb{F}_q$?

I found two related papers, There is a UGC hardness result here, https://www.cs.cmu.edu/~venkatg/pubs/papers/qaryCSP.pdf A kind of a stronger result might be found in these two other papers, http://...
4
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0answers
125 views

Is there a certificate definition for polyL?

Arora explained in his book a certificate definition for $\mathsf{NL}$ using a read-once tape. Can we apply a similar definition for the class $\mathsf{polyL}$?
3
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1answer
267 views

Is there any good literature on the computational complexity of function problems?

There are some cstheory questions that touches function-problems. Like this: Complexity class corresponding to sorting So here is the question: Is there good literature about the computational ...
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0answers
106 views

Two definitions of $QMA$

In this question, I am trying to understand the equivalence between the following two definitions of the complexity class QMA. In Quantum Computational Complexity, John Watrous defines the class QMA ...
5
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1answer
404 views

How is the VP=VNP question in char 2 different from other char? What is the current frontier in regards to this question?

What are the caveats one should be aware of when pursuing VP=VNP question in char 2 compared to other char? What is the current frontier in regards to this question?
14
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1answer
391 views

Problems in NC not known to lie in NC2

Are there interesting problems that are in $\mathsf{NC}$ but not known to be in $\mathsf{NC^{2}}$? In the paper 'A Taxonomy of Problems With Fast Parallel Algorithms', Cook mentions that MIS was known ...
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0answers
53 views

About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
2
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0answers
116 views

What would faster Fourier Transform(FFT?) and/or multiplication algorithms imply?

There are many problems which have implications on P vs. NP and other complexity classes. Supposing that we're interested in Fourier transforms and/or multiplication algorithms, do faster Fourier ...
2
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0answers
108 views

Are there analogues of Specker sequences for other complexity classes?

Consider the standard definition of computable real numbers: a real number $r$ is computable just in case $r$ is the limit of a sequence $(a_i)_{i \in \mathbb{N}}$ such that (1) the function $i \...
3
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0answers
147 views

The complexity of the disjoint union of a sequence of complete problems

Suppose that for every $k \in \mathbb{N}$ the decision problem $A_k$ is hard for $\mathsf{N}k\text{-}\mathsf{ExpTime}$. What is the complexity of their disjoint union $A = \{ (k,x) \mid k\in \mathbb{...
6
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1answer
308 views

Parity P and AM

What is known about non-trivial inclusions of $\oplus\mathsf{P}$ in other classes? In particular, is it known whether $\oplus\mathsf{P}$ is contained in $\mathsf{AM}$? The same questions apply to the ...
2
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0answers
160 views

About lower bounding the sample complexity of a distribution

Given a joint probability distribution over a finite number of random variables (each with a finite range space) of which only a certain subset is observable, is there a notion of "sample complexity" ...
2
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0answers
128 views

Complexity of counterexample function and bounded arithmetic

Let $\{L^c_i\}_{i}$ be an efficient enumeration of languages in $DTime(2^{n^c})$, e.g. clocked TMs. Assume $EXP\not = NEXP$. Let $L$ be an $NEXP$-complete language and therefore not in $EXP$. There ...
6
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1answer
482 views

Is “two or zero” matching in a bipartite graph NP complete?

I have this problem, which I haven't seen before in the literature: given a bipartite graph and a natural number $k$, can we select at least $k$ of the edges such that each left-hand vertex is ...
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0answers
141 views

Quantum computer versus Random 3-SAT?

It seems to be commonly believed that quantum computer cannot efficiently solve NP-hard problems. What about the challenging problems in average-case, such as Planted Clique and Random 3-SAT?
2
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0answers
193 views

Linear Time Hierarchy and Circuit complexity

By Karp-Lipton Theorem we have: $$PH\subseteq P/poly\Rightarrow PH=\Sigma^p_2$$ So this theorem suggest it is unlikely that $PH\subseteq P/poly$. I want to know is there any similar conditional or ...
19
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1answer
619 views

Division by two of functions in #P

Let $F$ be an integer valued function such that $2F$ is in $\#P$. Does it follow that $F$ is in $\#P$? Are there reasons to believe this is unlikely to always hold? Any references I should know ...
3
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1answer
226 views

Which is the approximation class of Minimum 0-1 integer programming if only non-negative integers are allowed?

I'm wondering which is the approximation class of Minimum 0-1 Integer Programming if only non-negative integers are used. That is: minimize $c^T \cdot x$, with $x\in \{0,1\}^n$ and $c\in (\mathbb{Z}^...
2
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1answer
389 views

Is there a problem currently known to be outside class $\mathsf{NP}\cup\mathsf{coNP}$ but inside $\mathsf{BPP}$?

May be this is trivial but I do not know the answer. As far as we know $$\mathsf{BPP}\subseteq\mathsf{\Sigma}_2\cap\mathsf{\Pi}_2$$ holds. As far as we know $$\mathsf{NP}\cup\mathsf{coNP}\subseteq\...
5
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2answers
190 views

Popular average-case complexity assumptions

Except for planted clique and random 3SAT, what are the other commonly used average-case complexity assumptions?
8
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1answer
663 views

Intersection of languages in NP

Can intersection of two languages in NP which are not NP complete be NP complete? Can intersection of two languages in coNP which are not coNP complete be coNP complete? Can intersection of two ...
3
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2answers
303 views

Consequences of $NP\subseteq P/poly$ to $BQP$

A post here Consequences of $BQP \subseteq P/poly$? queried on Consequences of $BQP \subseteq P/poly$. It is not known if $NP\subseteq BQP$. In general, what are the consequences of $NP\subseteq P/...
8
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2answers
261 views

Uncertainties in GCT program

In https://en.wikipedia.org/wiki/Geometric_complexity_theory it is mentioned that ".. Ketan Mulmuley believes the program, if viable, is likely to take about 100 years before it can settle the P vs. ...
6
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3answers
426 views

Natural NP-complete problems with high density?

(This question is related to a previous one, see the discussion in "Almost easy" NP-complete problems, but it may also be of independent interest, so I post it as a separate question.) Let ...
10
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1answer
502 views

Conjecture: All FPT NP-complete languages are fixed-parameter-isomorphic

Berman–Hartmanis conjecture: all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms[1]. I am interested in a more fine-grained ...
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1answer
312 views

Is this problem #P-hard and why?

Problem: In a directed graph $G=(V,E)$, each edge $e\in E$ is associated with a weight $w_e$ which is geometrically distributed with a parameter $p$, i.e. $P(w_e=i)=p(1-p)^{i-1}, i\geq 1$. $s,t$ are ...
3
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0answers
166 views

Big picture in counting complexities

(1) Is there a relation ( conjectured relation) between $\mathsf{\#P}$ and $\mathsf{CH}$? (2) How does $\tau$ conjecture in complexity of factorial fit in the picture? Is there a good reference? $\...
20
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1answer
688 views

“Almost easy” NP-complete problems

Let us say that a language $L$ is P-density-close if there is a polynomial time algorithm that correctly decides $L$ on almost all inputs. In other words, there is an $A\in$ P, such that $L\Delta A$...
18
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4answers
2k views

Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
8
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1answer
342 views

Are there more polynomial time problems with complexity lower bounds?

I'm looking for more problems in $P$ with classical time complexity lower bounds. Some people might wonder how you could prove such a lower bound. See below. Exponential Lower Bounds: Claim: If ...
16
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2answers
445 views

Potentially equal complexity classes without known contradictory relativizations

What are some examples of pairs of complexity classes $A$ and $B$ such that we do not know whether $A=B$, and we do not know contradictory relativizations either (i.e., we do not know oracles $P$ and ...
7
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1answer
258 views

Two DFA intersection emptiness connections to SETH & L vs P

(re "fine grained complexity") Wehar has proved that Two DFA intersection emptiness in $O(n^{2-\epsilon})$ time → SETH false. does anyone see any particular key proof difficulty, challenge, ...
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0answers
113 views

Complexity classes for problems that can be solved only from the length of the input

A tally language is a language on an alphabet with only one symbol. One can define complexity classes for tally languages, such as $P_1$ (the tally languages that can be decided in polynomial time). ...
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0answers
216 views

Is $\mathsf{NP}$ in $\mathsf{NNC}^1$?

Theorem 2.2 in "Nondeterministic circuits, space complexity and quasigroups", by Wolf, 1994 (a technical report version is available here without fee), proves that NP = NNC, where NNC is the class of ...
2
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0answers
91 views

Advances in complexity by studying particular problems

When we are trying to figure out in which complexity class a problem lies, we usually try simultaneously to come up with the best algorithm for it, together with the best hardness reduction, until (...
7
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0answers
290 views

Why is it so difficult to study Sum of Squares (SoS) algorithms with degree $d>4$?

In many publications on the computational complexity of Sum of Squares (SoS) algorithms, it is typical to study the degree-$4$ relaxation; e.g. Rounding Sum-of-Squares Relaxations Sum-of-Squares ...
8
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1answer
244 views

What's the complexity of counting odd nodes in graph?

According to Handshaking Lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number. This observation means that if we are ...
6
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0answers
136 views

Does it matter who begins communication in $IP(f(x))$?

Consider $IP(f(x))$, in other words, the class of languages that admit a private coin protocol $(P, V)$ running in $f(x)$ rounds (often in terms of the size of $x$), satisfying standard constraints. ...
3
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0answers
104 views

Number theoretic problems complete for $\mathsf{RL}$

Are there number theoretic problems (such as those related to $\mathsf{gcd}$) that are in $\mathsf{RL}$? Can these also be $\mathsf{RL}$-complete problems (is there any $\mathsf{RL}$-complete ...
23
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3answers
906 views

What are the relationships between those hypotheses in Fine-Grained Complexity Theory?

Complexity theory, through such concepts as NP-completeness, distinguishes between computational problems that have relatively efficient solutions and those that are intractable. "Fine-grained" ...
3
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0answers
61 views

Does simulating chiral gauge theories lie within BQP?

In theoretical physics, there is a branch of quantum field theory dealing with chiral gauge theories. It has been conjectured by Feynman [1] and others that all quantum field theories can be simulated ...
16
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1answer
400 views

Complexity classes for proofs of knowledge

Prompted by a question Greg Kuperberg asked me, I'm wondering if there are any papers that define and study complexity classes of languages admitting various kinds of proofs of knowledge. Classes ...
3
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1answer
278 views

Is counting the words in a finite regular language #P-complete?

Almost the exact same question was asked here, but nobody proved or cited its #P-completeness! I found this question because I proved it is #P-complete (proof below), and the proof was trivial, but I ...
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1answer
132 views

A curious statement in an old blog

In http://blog.computationalcomplexity.org/2009/08/finding-primes.html, a statement is added which reads "Oddly enough we would usually prefer a probabilistic over the deterministic method to find ...
-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 ...