Questions tagged [complexity-classes]

Computational complexity classes and their relations

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3
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1answer
49 views

Complete problem in $\Sigma_2^p$ - $\Sigma_{2}SAT$

It is known that the following problem is complete in $\Sigma_2^p$: $\Sigma_{2}SAT$ : Given a quantified boolean formula $\theta = \exists x_1,...,x_l\forall y_1,...,y_m\psi$, where $\psi$ is a ...
3
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1answer
153 views

complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
7
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1answer
181 views

What is the complexity class of higher-order primitive recursion?

Short and sweet: The complexity class of primitive recursive functions (WP) is PR (WP, Zoo). What's the complexity class of higher-order primitive recursion (WP)? The motivating context is simply that ...
1
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1answer
135 views

Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
0
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1answer
158 views

Lexicographic Boolean satisfiability

Maximal Satisfying Assignment (Lexicographic Boolean satisfiability / LexMaxSAT), the problem of finding the lexicographically maximum x_1, . . . , x_n ∈ {0, 1}^n that satisfies Boolean formula φ, or ...
3
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103 views

Complete problems for $(NP\cap CoNP)/poly$ class and universal representations

It is conjectured that $NP\cap CoNP$ does not have a complete problem with respect to the polynomial-time many-to-one reductions. I would like to know the current knowledge about the nonuniform ...
4
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0answers
189 views

What does width $4$ permutation branching program correspond to?

$L$ can be computed by a family of programs over $S_3$ of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of polynomial size. $L$ can be computed by a family ...
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195 views

Simultaneous evidence for $L\neq NL$ and $P\neq NP$

We believe $L\neq NL$ and $P\neq NP$. Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
2
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1answer
52 views

Relative error estimation of a special type of GapP function

Consider the functions included in the complexity class GapP. We know that approximating a function from GapP, in the worst case, to inverse polynomial multiplicative error, is #P-hard. Even correctly ...
1
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1answer
148 views

KRW Conjecture: separation of NC^1 and P

More than a real question this is a recap of something I have been studying. I hope someone will help me getting things straight, so any correction or thought about the following reasoning is more ...
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0answers
203 views

anything hinting that EXPTIME $\subseteqq$ PSPACE?

Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$?
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0answers
66 views

Additive error approximations of GapP functions

Consider a GapP function $g(x)$ for $x \in \{0, 1\}^{*}$. Consider an approximation $\tilde g(x)$ such that \begin{equation} \left|g(x) - \tilde g(x)\right| \leq \epsilon. \end{equation} Consider a ...
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0answers
181 views

IP, MIP and MIP* with super-polynomial verifier

Regarding each of the above classes, what are the currently known upper bounds when the verifier is given more than polynomial power? Specifically, when do we reach ALL in each of the above classes ...
3
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122 views

Complexity of real coefficients Linear Programs

I would like to know if there are known any polynomial time algorithms for deciding the feasibility of linear programs with real (not integers) coefficients. I know that for linear programs with ...
3
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101 views

Improving the approximation in Stockmeyer's counting theorem

Given a $\#P$ function $f(x)$, we can use Stockmeyer's counting theorem to get an approximation $g(x)$ such that \begin{equation} \left(1 - \frac{1}{\text{poly}(n)} \right) f(x) \le g(x) \le \left(...
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0answers
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Consequences of an $o(\log n)$-approximation algorithm for a $\mathsf{\log\text{-}APX}$ hard problem

In [1], Feige proves that if there is a polynomial algorithm with approximation ratio in $o(\log n)$ for any $\mathsf{log\textit{-}APX}$ hard problem (say Minimum Dominating Set), then $\mathsf{NP}\...
3
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1answer
127 views

What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0[2]$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
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On Determinant modulo $2^k$ complexity

Determinant of integer matrix modulo $2$ is complete for the class $\oplus L$. Is determinant modulo $2^k$ computable in $\oplus L$ at any fixed $k$? How about if $k=o(n)$ where matrix is $n\times n$?
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The graph of problem reductions

A classical approach to study the complexity of a problem $P$ is to efficiently reduce a well known problem $P'$ to $P$, thus showing that $P$ is at least as difficult as $P'$. The TCS literature ...
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1answer
142 views

Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
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73 views

What is the best reduction we know from flavors of $SAT$ to $MCSP$?

Consider $\mathcal P(n,m)$ to be a class from the set $\{k\mbox{-}\mathsf{SAT}(n,m),\mathsf{CIRCUITSAT}(n,m)\}$ where $k$ is fixed, $n$ is number of variables and $m$ is number of clauses. Denote $\...
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52 views

Is anything known about NC$^1$ with NP oracle

A few things are known about the class $\textsf{L}$ provided with an $\textsf{NP}$ oracle ($\textsf{L}^\textsf{NP} = \Theta_2^\textsf{P}$ has attracted a bit of attention, for instance [1]) On the ...
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0answers
51 views

Logarithmic queries to $\Sigma_i^P$ oracle and the Boolean hiearchies

If I understood correctly (the complexity zoo, wikipedia, and some of the cited articles), the class $\textsf{P}^{\textsf{NP}[\log]}$, also known as $\Theta_2^{\textsf{P}}$, sits at the top of the ...
5
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1answer
216 views

Where is $MA$ more relevant than $\exists BPP$?

(EDITED) A previous version of this question asked about a complexity class I called $MA^*$, which has been recognized by users to be $\exists BPP$. The difference between $MA$ and $\exists BPP$ is ...
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$\mathsf{ACC}^0$ and $\mathsf{TC}^0$ with $\mathsf{Cuniform}$-$\oplus\mathsf{L}$ or $\mathsf{Cuniform}$-$\mathsf{NC}^1$ oracle?

$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it. Following inclusions are known: $$\mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\...
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1answer
115 views

Is there any relation between $\mathsf{PPP}\subseteq\mathsf{TFNP}$ and $\mathsf{UP}$? [closed]

A complete problem for $\mathsf{PPP}$ is the pigeon problem which is 'Given a Boolean circuit ${\displaystyle C}$ having the same number ${\displaystyle n}$ of input bits as output bits, find either ...
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1answer
249 views

Is there a containment between $\mathsf{ZPP}$ and $\mathsf{UP}$?

Do we know if $\mathsf{ZPP}\subseteq\mathsf{UP}$ known or is there oracles against the hypothesis?
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2answers
392 views

NLOGTIME versus $\exists$DLOGTIME

$\def\dlt{\mathrm{DLOGTIME}}\def\nlt{\mathrm{NLOGTIME}}\def\mr{\mathrm}$During a recent discussion on another question, I mentioned a factoid $\exists\dlt=\nlt$, but then I realized that I may have ...
12
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340 views

NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
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73 views

Is it to solid to conclude APX-complete after showing a problem cannot be approximated better than 1.5 and also develop a 2-approximation algorithm

I know the canonical way to show APX-Complete is to give an L-reduction from an already-known APX-complete problem. If I have used gap-preserving reduction to show a problem cannot be approximated ...
2
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1answer
146 views

What graphs on $\mathbb{N}$ can be encoded as regular languages?

Suppose I represent the natural number 0 by "x", and use the symbol "s" for successor so that I get the following encoding of $\alpha : \mathbb{N} \rightarrow V$ of natural numbers ...
19
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1answer
1k views

Has parameterized complexity led to better algorithms?

I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...
2
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1answer
170 views

Complexity of type inference in the simply typed lambda calculus

A similar question was answered here: Is simply typed lambda calculus equivalent to primitive recursive functions What I conclude from the answers is that the complexity is that of the extended ...
1
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1answer
80 views

Rademacher Complexity of the Composition with an Indicator

Consider the statistical learning setting where you have an arbitrary hypothesis space $\mathcal{H}$, a data space $\mathcal{Z}$, and a bounded loss function $\ell: \mathcal{H}\times \mathcal{Z} \...
1
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1answer
86 views

Can a NEXP machine simulate invalid queries to a promise problem oracle?

Let $A=(A_{YES},A_{NO})$ be some promise problem (such as xSAT, the Local Hamiltonian problem, etc). Suppose we want to show that a P machine with access to a the oracle A can always have its output ...
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0answers
64 views

Reference to "compressibility" of logarithmic space [closed]

Is there a reference somewhere for the result SPACE($O(\log n)$) = SPACE($\log n$)? i.e. Big-O doesn't matter in logspace since you can compress the space. I feel like this is an elementary result but ...
1
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1answer
91 views

Consequences of turning $\oplus \text{SAT}$ into few satisfying assignments

Suppose there is a reduction which, given a $\oplus \text{SAT}$ instance $\phi$, returns another $\oplus \text{SAT}$ instance $\psi$ having all the following properties: The size of $\psi$ is ...
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Counting on grid graphs

Are there problems defined on graphs, such as counting 2-factors, Hamiltonian cycles, connected spanning subgraphs etc., that are in $\#P$ and remain hard for grid graphs? Since there seem to be ...
6
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1answer
142 views

Complexity of approximating a real function using queries

Consider the following computational problem, where $I$ is the real interval $[-1,1]$: There is a monotonically-increasing function $f: I\to I$. You are allowed to access it only through queries of ...
-5
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1answer
137 views

Is mathematical proof itself NP-hard?

At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
3
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1answer
142 views

Are all RegExp solvable in O(n)?

I'm wondering if all features, that are often part of modern RegEx engines, are solvable in O(n). I'm talking about features like repeating patterns ([abc]+);\1 ...
1
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1answer
131 views

Diagonalization arguments for QMA type proof systems

Diagonalization is a very common technique to find oracle separations. For example, it can be used to separate $\cal{P}$ and $\cal{NP}$, with the essential idea being that of constructing an oracle in ...
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0answers
145 views

Why can MIP be restricted to just two provers?

In several places I see it referred to that the MIP class can be assumed to be two interactive provers that don't communicate with each other, rather than any polynomial number of provers. Why are ...
12
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2answers
289 views

Is there an assumption that implies $P=ZPP$ which is not known to imply $P=BPP$?

There are assumptions that are known to imply that $P = BPP$. For example, if there exists a function in $E = DTIME(2^{O(n)})$ that has circuit complexity $2^{\Omega(n)}$, then $P = BPP$ [1]. Clearly, ...
7
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1answer
186 views

Complexity class of efficient streaming algorithms

Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words: $L \in \mathsf{StreamL}$ if there ...
5
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0answers
124 views

Is there an analogue of QMA where Merlin gives Arthur unitaries rather than states?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$Is there an analogue to $\mathsf{QMA}$ where Merlin provides to Arthur single-use access to a unitary operator $U$? By ...
10
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1answer
307 views

The theoretical complexity of Go - The state of the art

What are the latest advances in theoretical complexity of Go? I know some early works about the complexity of Go: "Go is polynomial-space hard" proved that Go is PSPACE-hard. "Ladders are PSPACE-...
2
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0answers
101 views

Is PP invariant under changing its cut-off from 1/2 to another number?

Suppose I have a fixed family of quantum circuits $\{C_i\}$ for which determining whether the maximum output acceptance probabilities are $p\geq 1/2$ or $p< 1/2$ is PP-hard. Now suppose I have the ...
0
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2answers
117 views

Order of quantifiers in the definition of NP-completeness: does the reduction allow arbitrary polynomials? [closed]

Arora and Barak define NP-completeness as the following: "We say that a language $A \subseteq \{0, 1\}^∗$ is polynomial-time Karp reducible to a language $B \subseteq \{0, 1\}^∗$ denoted by $A \leq_p ...
6
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1answer
162 views

Is $L \subset 1NL$ when $L \neq NL$?

A log-space Turing machine has a read-only input tape, a write-only output tape and uses at most $O(\log n)$ space in its read-write work tapes. The classes $L$ and $NL$ contain those languages which ...

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