Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
2,564 questions
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1answer
34 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 ...
-1
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0answers
11 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 ...
0
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0answers
26 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 ...
15
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1answer
233 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 ...
1
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0answers
49 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
votes
0answers
118 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
39 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 ...
7
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3answers
318 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
votes
1answer
116 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
votes
1answer
114 views
Which (almost) balanced Boolean function has smallest “total” influence
The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f : \{-1,1\}^n \xrightarrow{} \{-1,1\}$
$$
\max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot \...
8
votes
1answer
187 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
votes
1answer
79 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
votes
1answer
64 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
109 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
278 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
342 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
votes
0answers
111 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 ...
10
votes
1answer
339 views
+150
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
150 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
vote
0answers
71 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
votes
0answers
81 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
vote
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
votes
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
vote
1answer
57 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 ...
1
vote
0answers
109 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 ...
1
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0answers
75 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
votes
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
67 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
votes
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
votes
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
votes
0answers
100 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 ...
1
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0answers
76 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}=...
1
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0answers
35 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
vote
2answers
102 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
67 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 \...
0
votes
1answer
66 views
Question on deduction that a certain problem requires exponential space
My question concern's a statement from the classic paper The equivalence problem for regular expressions with squaring requires exponential space.
Regular expressions with squaring are like ordinary ...
8
votes
1answer
208 views
Depth reduction for Boolean circuits
This result by Tavenas, Koiran and others show that any polynomial computed by a circuit of size $s$ is computed by a depth-4 homogenous circuit of size $s^{\sqrt{d}}$.
Are there any similar results ...
6
votes
1answer
227 views
Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)
In the wikipedia article on Time Complexity it is written that:
The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
1
vote
0answers
243 views
On $BPP$ in $P^{NP}$ and $SETH$
It is believed showing $BPP$ in $P$ involves good $PRG$s and faces lower bound barriers.
Does showing $BPP$ in $P^{NP}$ which would mean $BPP\neq EXP^{NP}$ face similar $PRG$ and give lower bounds?
...
9
votes
1answer
243 views
What is the reference for the proof Gödel's first incompleteness theorem based on the undecidability of the halting problem?
A weaker form of Gödel's First Incompleteness Theorem, direct proofs of which in Gödel's manner are lengthy, involved and at some place rather counter-intuitive, has a simple and intuitive proof based ...
0
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0answers
67 views
Where is the flaw in this proof that an LP solves TSP? [duplicate]
In this preprint on Arxiv, M. Diaby, M.H. Karwan, and L. Sun give a Linear Program which they claim solves the Traveling Salesman Problem. In contrast to their prior work, which was asked about here, ...
3
votes
1answer
89 views
Computational hardness for sampling a uniform matching
A famous result of Jerrum, Sinclair, and Vigoda shows that there exists a polynomial-time algorithm which takes a bipartite graph $G$ and produces a random perfect matching $M$ of $G$ (assuming one ...
6
votes
0answers
147 views
Can reciprocal inputs speed up monotone computations?
A $(+,\times,1/x_i)$ circuit is a standard monotone arithmetic $(+,\times)$ circuit with the only difference that now besides the input variables $x_1,\ldots,x_n$, also their reciprocals $1/x_1,\...
2
votes
1answer
143 views
An equation relating Time complexity, Space complexity, and entropy of output
Is there an equation that relates minimum time complexity, minimum space complexity, and entropy of the output of a function? It seems to me that there should be a relatively intuitive relationship ...
-2
votes
1answer
115 views
Does P^NP=NP imply NP=coNP? [closed]
If you have it, the proof would be appreciated.
Note: P^NP means P with NP oracle
2
votes
0answers
76 views
Best algorithms for real linear programming
Linear Programming asks for $x\in\mathbb R^n$ such that $Ax\leq L$ holds where $A\in\mathbb R^{m\times n}$ and $L\in\mathbb R^m$ are given. Karmarkar has shown that $\ell$ is the number of bits of ...
3
votes
0answers
68 views
Complexity of counting Wang tiles
Consider the question of counting Wang tilings on a torus. The decision version of this problem is known to be NP-complete. Is the counting version #P-complete?
