All Questions
1,809 questions
5
votes
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540
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Minimal Backtracking Proof Tree
When trying to prove that a particular instance of a problem like graph coloring or SAT is unsatisfiable, generally one explores the search tree using an algorithm like DPLL and the proof of ...
- 601
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1
answer
176
views
Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 151
5
votes
1
answer
138
views
Complexity and length
Suppose we define continuous piecewise linear functions $f$ on $[0,1]$ using your favorite programming language, or by finite automata, or by any other suitable machine. Define the complexity $H(f)$ ...
- 2,934
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1
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462
views
Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
- 1,462
5
votes
1
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315
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On optimal dual solutions for the minimum weight perfect matching problems in the case of metric weights
Following Lovasz-Plummer (Matching theory, North-Holland 1986, Theorem 9.2.1),
the minimum weight perfect matching problem on a complete graph
$G$ with even number of vertices and weight $w:E(G)\to
\...
- 4,878
5
votes
1
answer
424
views
What is the LP gap of vertex cover in planar graphs?
What is the LP gap of vertex cover in planar graphs?
The LP I refer to is min $\sum_{e \in E } c_e x_e \ \ $ subject to $ \ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E $
$ c_e \geq 0 $ are ...
- 111
5
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1
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84
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a question about complexity of Boolean problem
I was thinking in solving the following problem for the general case :
**) Given a list of pairs $((n_i, A_i))_{i=1}^k$, where for each $i$ we have that $n_i $ is a non-negative integer, and $A_i$ is ...
user95470
5
votes
1
answer
146
views
How does one go from convexity to submodularity?
If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...
- 1,893
5
votes
1
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327
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Subsets of all Diophantine's sets
I have asked this question on math.stackexchange already:
https://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets
Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable $\...
- 2,576
5
votes
1
answer
301
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NP-hardness of sparsest cut
Consider bipartitioning the vertices of a graph $(V,E)$ into $V = P \cup Q$ to minimize $$\frac{|E(P,Q)|}{|P| |Q|},$$ where $E(P,Q)$ denotes the set of edges in the cut. The usual citation for NP-...
- 51
5
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1
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3k
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Maximizing supermodular functions
I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a ...
5
votes
2
answers
888
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relation between solution of a linear program and its perturbation
I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$:
$$
\max_j c' x_j
$$
Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...
- 373
5
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0
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75
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What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
5
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0
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199
views
In finite model theory, is "invariant FOL with $\varepsilon$-operator" unavoidably second-order?
Throughout, all structures are finite.
Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary iff it is the class of finite models of a sentence in the ...
- 21.2k
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0
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192
views
Complexity implications on computability
Are there any known links between complexity theory and computability theory by which I mean non-trivial theorems of the form: If NP $\neq$ co-NP then there is no strong minimal pair of r.e. sets or ...
- 3,029
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0
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184
views
What is the fastest algorithm for multiplying one given number with many others?
When multiplying two numbers with each other, which are $n$-bit numbers, there are several algorithms like the one of Karatsuba ($O(n^{\log_2 3})$) and a new one doing it even better (Harvey - Van der ...
- 749
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0
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180
views
Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
- 151
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0
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129
views
Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics
Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...
- 1,833
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0
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292
views
Fastest sine of a large power of 2
What is the fastest known way to calculate $\sin(2^{n})$ for large integer $n$?
I only need the highest few bits to be correct. I suspect that the compute time required
scales with $n$ (and actually ...
- 1,547
5
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0
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180
views
Certificate for computation of ideal class group
Is there a known way of producing a certificate that can be used to more quickly verify that an ideal class group of a number field was computed correctly? More formally, I would like to know if there'...
- 1,856
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240
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Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
- 123
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267
views
Are there any neusis-hard/neusis-complete problems?
I have lately been enjoying Richeson's Tales of Impossibility (see MAA review), an accessible book on the famous problems of Euclidean geometry including angle trisection/cube doubling/heptagon ...
- 2,185
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246
views
Does $\mathsf{Q}$ have any interesting provably recursive functions?
This question was asked and bountied at MSE without success.
For an appropriate theory $T$, say that an $n$-ary $T$-provably recursive function is a $\Sigma_1$ formula $\varphi$ with $n+1$ free ...
- 21.2k
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0
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85
views
special classes of ideals (eg. toric) that admit faster Buchberger algorithm?
I have heard that toric ideals allow one to speed up the Buchberger algorithm considerably (see Grobner bases of toric ideals, Remark 2,3). My question is two-fold:
What are the precise complexity-...
- 1,221
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301
views
The expressiveness of functions computable on trees
Motivation:
Let's define a function computable on a $k$-ary tree as a function composed with simpler computable functions defined at each node such that a function of this kind defined on a binary ...
- 3,871
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0
answers
125
views
Is integer circuit membership undecidable?
According to wikipedia
integer circuit
in its simplest form is succinct representation of multivariate polynomial with
integer coefficients. Decidability if an integer is represented by the integer ...
- 25.4k
5
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87
views
Problem to efficiently compute the Volume of $d$ anchored 4D cuboids
An easy still unsolved special case of Klee's measure problem with applications in multiple objective optimization is described in the following.
Let $[\vec a_1,\vec b_1],\dots,[\vec a_n,\vec b_n]$ ...
- 4,535
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0
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78
views
Complexity of calculating $f^{(n)}(0)$/extracting a coefficient of a functions taylor-series
Many combinatorial problems can be solved using generating functions.
In such a case, we obtain a function $f(x)$, which (for usual) has a taylor-expansion:
$$
f(x) = \sum_{n\ge 0 } a_n x^n
$$
So ...
- 151
5
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0
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265
views
Finite axiomatizability and $\mathrm{PA^{top}}$
Is $\mathrm{PA^{top}}$ finitely axiomatizable? If not, does it have a finitely axiomatizable extension (allowing new predicates but not new variable types) that has arbitrarily large finite models?
$\...
- 7,545
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162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
- 1,150
5
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0
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137
views
How many CNOT gates are needed to compute scalar multiplication in a finite field of characteristic 2?
I want to know if scalar multiplication over a finite field of characteristic 2 is generally easier to compute than a linear transformation over a vector space over $F_{2}$.
Suppose that $p(x)$ is an ...
5
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0
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106
views
Collapsing the Exponential time Hierarchy with a complete language as oracle
It is known that $\mathsf{P^A=NP^A}$ is true for every $\mathsf{EXP}$ complete language $\mathsf{A}$. The question is the whether the similar things hold for Exponential time Hierarchy.
Is there ...
- 1,689
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0
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307
views
Quantum P vs NP equivalent problem
If $P = NP$, does it follow that $BQP = NP^{BQP}$?
I came up with this question when I was thinking about how $P = NP$ can be described as "does every decision problem where a proof for YES can be ...
- 51
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0
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84
views
One-sided version of the random oracle hypothesis
The random oracle hypothesis states that relationships between complexity classes hold iff they hold relativized to a random oracle with probability 1. This is false, but all of the counterexamples I ...
- 2,130
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0
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126
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Collapsing the Intuitionistic Bounded Arithmetics Hierarchy
Let $iT$ be the intuitionistic first order theory with non-logical axioms of classical first order theory $T$.
Theorem1. If $\mathsf{T^i_2}\vdash \mathsf{T_2}$, then $\mathsf{T^i_2}$ proves that the ...
- 1,689
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0
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166
views
Fourier basis for sub-Gaussian spaces?
Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with ...
- 519
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145
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Complexity of $\mathbb{Z}^n$ tilings
Let $\mathcal{T} \subset \mathbb{Z}^n$ be a finite set. Let $\Lambda \subset \mathbb{Z}^n$ be a full rank lattice.
We say that $\mathcal{T}$ is a $\Lambda$-tile for $\mathbb{Z}^n$ if the following ...
- 800
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222
views
Littlewood-Richardson rule for the complete flag variety: GapP complete?
The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
- 2,168
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240
views
Complexity of approximating the size of the range of a matrix
Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...
- 3,377
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0
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263
views
Is there a program for theory of incompleteness in NP?
Motivated by Suresh's post, Techniques for showing that problem is in hardness limbo, it seems that there might be an underlying theory that explains why some of these problems can not be complete for ...
- 2,993
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0
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167
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A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
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617
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Any reason to believe that $NP \neq P$ is unprovable in ZFC
We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...
- 3,094
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0
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194
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A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
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171
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Computational complexity of resolution of singularities of varieties over fields with characteristic 0 [closed]
What is the computational complexity of resolution of singularities of varieties over fields with characteristics 0?
- 3,094
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0
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139
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Are there sampNP-intermediate problems?
This questions is approximately cross-posted from theoretical computer science stackexchange
Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\mathsf{NPI} := \mathsf{NP} \...
- 1,368
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0
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182
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Best lower bound for proof complexity of graph asymmetry
Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the ...
- 2,993
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0
answers
204
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A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
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0
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682
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Is integer factorization harder than RSA ($n=pq$) factorization? [closed]
This is a repost. I could not get a precise answer on math.SE and cstheory.SE
Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $...
- 236
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0
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581
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When is polytope compatible with network flow?
A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
- 3,475
4
votes
3
answers
767
views
Does NP = "epsilon-P" (PTAS / BPP)?
Some NP-complete optimization problems, like the knapsack problem, have a solution reachable in polynomial time that is guaranteed to be within arbitrary ε of the optimum answer. (aka PTAS - ...
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