Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
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72 views
Is it possible to sort by only knowing the sign of pairwise sums?
I am currently thinking of how much structure one actually needs in order to be able to sort things at all. All comparison-based algorithms need a direct comparability, but are we able to remove this ...
2
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0answers
79 views
Is $MSB$ of permanent and certifying half number of witnesses easy?
Can there be a $P$ algorithm to decide if number of perfect matchings is at least $(n!/2)+1$ for a bipartite graph on $n+n$ vertices?
Can there be a $P$ algorithm to decide if number of witnesses ...
2
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0answers
33 views
Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles
Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles.
Question: Is there some constant $k$ so that the $...
1
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1answer
92 views
Count satisfying assignments of CNF formulas over all possible negation assignments
Consider the set of all CNF instances that can be generated by adding negations to a single monotone CNF instance. How hard is it to compute the sum of the counts of satisfying assignments for the set?...
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0answers
105 views
$n^2 x n^2$ Sudoku Puzzles have equal amount of easy-instances that can be solved in $O(n^k)$ time, under experimentation so far this is correct
Introduction
I found out that I can take any $n^2 x n^2$ puzzle, solve it and study its permutation pattern and implement it into a poly-time Sudoku Generator. I also found out that a limited subset ...
4
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0answers
103 views
Planar Exact Cover by even-size sets
Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one.
Is the following problem NP-complete?
Planar Exact Cover by even-size sets
Input: A set $U$, a ...
3
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0answers
96 views
Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power
It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$.
For example, authors of this paper say that
... any constant
depth circuit ...
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0answers
51 views
Graph automorphism with prescribed values
Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
4
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2answers
746 views
Attempted proofs of P vs NP
What are the most recent (say in the last 3 years) attempts at disproving $P = NP$, and where can I find the papers?
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0answers
61 views
Validity of relativization argument
In the book 'Computational complexity' by Arora-Barak, I was studying that any result of TM/ complexity classes that use only the following two properties of TM's are called relativizing results :
$I....
1
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0answers
119 views
Can a hash preimage be used to amplify BPP probabilities?
Suppose we have an algorithm in $\mathrm{BPP}$ to decide membership in a language $L$. Normally we amplify the probability of accepting and rejecting members correctly by running our algorithm ...
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1answer
106 views
Is the unbounded fan-in model realistic?
Does the unbounded fan-in circuit model apply in "practical" settings? In other words, are there real-world realisable computers with unbounded fan-in gates?
As I understand, standard silicon ASICs ...
-1
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1answer
84 views
Comprehensive list of functions used in Big-$O$ notation
We all know that exponential functions grow faster than polynomials. Let us consider the following function: $f(n)=n^{a_1}⋅(\log n)^{a_2}⋅(\log\log n)^{a_3}⋅(\log\log\log n)^{a_4}⋯ $ where the leading ...
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0answers
17 views
How to deal with tradeoffs while making choices?
Sometimes while writing some code I make some decisions, the "problem" is that every choice has a negative consequence. Is coming with the perfect architecture possible? You'll tell me; perfect ...
2
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2answers
190 views
Under what models do we know linear time sorting?
The best we know for general case sorting is $O(n\log n)$ (which is also $\theta(n\log n)$ is decision tree model) and the problem of $O(n)$ sorting is open for turing machine models.
Under what ...
20
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1answer
391 views
Is prime-counting function #P-complete?
Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
2
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0answers
56 views
$XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?
Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following:
$FPT = XPuniform\implies EPTAS = PTAS$.
Fundamentals of Parametrized complexity on page $534$ has
...
2
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0answers
131 views
Is the following problem in $coNP$?
Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$?
At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
0
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0answers
43 views
What is the complexity of Parametric Mixed Integer Linear Programming?
We know
$$\forall\bf y\in\mathbb Z^n:K\bf y\leq b$$
$$\exists\bf x\in\mathbb Z^m:A\bf x + B\bf y\leq c$$
is in $\bf P$ if $n,m$ are fixed from Kannan's result (refer page $1$ in reference).
What is ...
8
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3answers
388 views
Why exactly are complexity theorists interested in closed timelike curves?
Context:
There are several papers that study the implications of closed timelike curves (CTCs) to quantum complexity. In 2008, Aaronson and Watrous published their famous paper on this topic which ...
2
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1answer
117 views
Lower bound on alternations needed in $BQP$ versus $PH$ result?
What is the fastest $f(n)$ the relatively new result of oracle separation of $\mathsf{BQP}$ from $\mathsf{PH}$ provides such that ${\#\mathsf{SAT}}\not\subseteq\mathsf{FP}^{\mathsf{PH}[O(f(n))]}$ ...
4
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1answer
164 views
Which (almost) balanced Boolean function has smallest “total” influence
The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f\colon \{-1,1\}^n \xrightarrow{} \{-1,1\}$
$$
\max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot ...
8
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1answer
251 views
Is there any quantum analog of the VP vs. VNP problem?
From Wikipedia:
$\mathsf{VP}$: The class VP is the algebraic analog of P; it is the class of polynomials $f$ of polynomial degree that have polynomial size circuits over a fixed field $K$.
$\...
-1
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1answer
80 views
Why is $BPP^{NP}$ in polynomial hierarchy? [closed]
Why is $BPP^{NP}$ in the polynomial hierarchy?
I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
2
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1answer
65 views
Hardness of LWE on not-uniform vector samples
The "usual decisional LWE":
The challenger and the adversary get a common random matrix $A \in F_{q}^{m \times n}$. The challenger chooses a secret $s \in F_{q}^{n}$ and generates random (small) ...
2
votes
1answer
112 views
How to use a 𝑝-coin so a TM can decide an undecidable language in polynomial time? [closed]
In "Computational complexity- A modern approach" book (page 117) for the lemma 7.12 (following) the author mentioned that if the ρ is efficiently computable ρ-coin cannot give probabilistic algorithm ...
10
votes
1answer
285 views
How hard is deciding the existence of Red-Blue perfect matching?
Two-colorable perfect matching problem is to decide whether a graph has coloring with two colors such that each node has exactly one neighbor the same color as itself. The problem was proven to be NP-...
10
votes
1answer
349 views
An obstruction like ETH
We know under $ETH$ we cannot solve $K$-SUM in $f(K)poly(nK)$ time under any function $f(K)$ (usually $2^{O(K)}$).
Is there any conjecture that prevents a $(\log n)^{O(K)}$ complexity (this is ...
2
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0answers
112 views
A complexity-class of problems that cannot be solved in finite time
Consider the following game: Alice chooses a real function number $x\in [0,1]$; Bob has to guess the number by asking Alice any number of queries of the form "is $x > a$?" [where Bob can choose ...
11
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1answer
540 views
Technical issue with PCP theorem proof
I am reading the proof from here and I stumbled upon a technical (yet crucial) problem. I know this is rather specific and the context is problematic, but I couldn't figure it out myself.
In pages 51 ...
11
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0answers
153 views
$\exists \mathbb R$ and IP
We know NP$\subseteq$ $\exists \mathbb R$$\subseteq$ PSPACE=IP, but is there some more direct proof for $\exists \mathbb R\subseteq$ IP?
What about the other direction, are there some Arthur-Merlin ...
1
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0answers
72 views
Minimum cut with nonlinear objective function
Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum.
Let us generalize it the following way: let $f$ be a ...
3
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0answers
82 views
Counting quotient graphs, but not exactly
All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
1
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1answer
142 views
Does Hadwiger conjecture imply that NP = coNP?
(Disclaimer: I suspect the answer is no, but I fail to see why)
Here is a nice picture by David Epstein (taken from Wikipedia) illustrating Hadwiger's conjecture:
The point is that if in a given ...
2
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0answers
48 views
Complexity of comparing extended integer power towers
Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
1
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1answer
59 views
Complexity status of the Edge Deletion problem to bounded degree graphs
I'm interested in the complexity status of the following problem.
Input: a graph $G=(V,E)$ and two natural numbers $k$ and $d$.
Output: Yes, if there exists a subset $E' \subseteq E$ of cardinality ...
3
votes
1answer
130 views
How small can extension complexity be?
In this article on extension complexity of regular polygons https://arxiv.org/pdf/1505.08031.pdf it is mentioned that extension complexity of $n$ regular polygons should be $\theta(\log n)$. This is ...
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0answers
111 views
Canonical complete problem for $\mathrm{FP}^{\Sigma^p_2}$
Given a $\Sigma^p_2$-complete oracle (i.e., $\Sigma_2 \mathrm{SAT}$), I have a problem that requires to call this oracle polynomially many times and returns an integer. Essentially, this is a function ...
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0answers
76 views
Entropy bounds on solutions to problems in BPP and other complexity classes based on entropy demands
Has anyone studied the asymptotics of problems in complexity classes like $BPP$? The thought came to me that if a problem in $BPP$ only requires $O(log(n))$ bits of entropy to solve then, intuitively, ...
4
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0answers
96 views
Arranging letters to make a word in a regular language
Fix a regular language $L$ on the alphabet $\{a, b\}$, and consider the following problem. I am given as input:
some number $m \in \mathbb{N}$ of copies of the letter $a$, and
some number $n \in \...
2
votes
0answers
68 views
how is time complexity defined in computational learning theory
In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$.
Now ...
3
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0answers
135 views
Testing emptiness property complexity in Sum of Squares Proof systems
Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where $$h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
3
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0answers
53 views
Complexity of #PP2DNF where we also count on the number of clauses
The #PP2DNF problem is the following: we have variables $X = \{x_1, \ldots, x_n\}$, $Y = \{y_1, \ldots, y_n\}$, and a positive partitioned 2-DNF formula, i.e., a Boolean formula of the form $\phi = \...
8
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0answers
104 views
Can we define a meaningful concept of exptime reductions (as opposed to polytime reductions) for classes like NEXP or NEEXP?
Typically we are only interested in polytime reductions as we are usually interested in showing a reduction from one NP-problem to another.
However, if we consider larger complexity classes such as ...
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0answers
94 views
Complexity of planted root of a system of quadratic homogeneous polynomials?
Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
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0answers
37 views
Problems rephrased as quadratic unconstrained binary optimization
I was impressed when i came across Quadratic unconstrained binary optimization (QUBO) recently, and saw how one can rephrase many combinatorial problems into questions about optima of binary functions....
3
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0answers
145 views
Take a NEXP-complete problem and then have the input in unary. Why is this not NP-complete?
It is known that if any unary language is NP-complete, then P=NP.
Suppose we take a NEXP-complete language with input $x$ in binary and witness $y\in\{0,1\}^{2^{poly(|x|)}}$ such that the verifying ...
11
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0answers
175 views
Computational Complexity of the Frobenius Problem
The Frobenius problem takes as input $n$ positive integers $a_1,\ldots,a_n$ with $\gcd(a_1,\ldots,a_n)=1$ and asks for the largest integer $F$ that cannot be written in the form $F=a_1x_1+a_2x_2+\...
1
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2answers
103 views
When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]
In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
4
votes
0answers
70 views
Refinement of the hierarchy theorem of a complexity class
Say $\mathrm{CTIME}$ is some complexity measure, syntactic or semantic (e.g. $\mathrm{DTIME}$ or $\mathrm{BPTIME}$).
If we already know that for some $f(n) = \omega(n)$, $\mathrm{CTIME}(n) \subsetneq \...
