**2**

votes

**0**answers

55 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

**24**

votes

**1**answer

717 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

**7**

votes

**1**answer

418 views

### Can we invert barycentric subdivision?

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question:
(1) Let's say we know that a finite simplicial complex $S$ is the ...

**7**

votes

**1**answer

447 views

### Normality of Chaitin's constant

Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition?
Even if we replace the existential quantifiers in the assertion of non-normality ...

**12**

votes

**0**answers

320 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

**1**

vote

**0**answers

93 views

### Schönhage's SMM with only one instruction

It is possible to implement $\lambda$-calculus in Schönhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...

**4**

votes

**2**answers

247 views

### Hardness of approximation of Dominating Set

It is stated throughout the computational complexity literature that the Dominating Set problem is NP-hard to approximate within a factor of $\Omega(\log n)$. To my knowledge, the first and only proof ...

**10**

votes

**4**answers

929 views

### Quantum algorithms for dummies

I want to try my hand at designing quantum algorithms to solve certain problems. I feel like I understand (for example) how Grover's algorithm and Shor's algorithm work, and I'm excited to apply the ...

**1**

vote

**1**answer

199 views

### Longest run of heads

Consider $n$ independent tosses of a fair coin; the sample space has $2^n$ elements. Let $R_n(x)$ be the length of the longest run of heads in outcome $x$. We know that $$E[R_n]=\Theta (\log n)$$
...

**1**

vote

**0**answers

81 views

### Are set covering problems with nonlinear cost functions NP-Hard?

Are set covering problems (set cover problem wikipedia) with a nonlinear cost function also NP-hard? Is there a general result about this?
To be more specific the cost function I am interested in ...

**1**

vote

**1**answer

168 views

### Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

I failed to get an answer at http://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In ...

**12**

votes

**3**answers

2k views

### Will quantum computing kill cryptography ? [closed]

I apologize as this question is not really mathematical, and therefore perhaps not
well-suited for this site. Please feel free to close it if you think it is not. My reason
for asking it here is that ...

**1**

vote

**1**answer

109 views

### Separation of Anti-Hole Inequality

Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...

**5**

votes

**0**answers

129 views

### 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?

**6**

votes

**2**answers

772 views

### Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...

**6**

votes

**2**answers

608 views

### Strassen's algorithm

I am reading Landsberg's "Tensors: Geometry and Applications". Here he mentions tensor formulation of Strassen's algorithm and shows that the rank of Strassen's matrix multiplication tensor is $7$ and ...

**0**

votes

**0**answers

59 views

### Approximation for accumulative set cover

Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\{1,\ldots,m\}$, I want to maximize the quantity
\begin{equation}
...

**5**

votes

**1**answer

282 views

### Maximal chain of 1s in binary strings

Let $S$ be the set of $2^n$ binary $n$-bit strings. For every $x\in S$, let $f(x)$ is the maximal chain of bits 1 in $x$. So Can we find a good upper bound of $$F(n)=\frac{\sum_{x\in S}f(x)}{2^n}$$
Of ...

**18**

votes

**0**answers

505 views

### Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...

**1**

vote

**1**answer

222 views

### definition of NP [closed]

"Popular" discussions of PvsNP often begin by characterizing NP problems as those for which a purported solution can be verified in polynomial "time". Later the definition changes to; the problem is ...

**3**

votes

**1**answer

216 views

### The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...

**2**

votes

**2**answers

134 views

### Size-limited oracles

I am interested in complexity of algorithms which have access to the following peculiar sort of oracle:
Suppose that an invocation of an algorithm f with an input of size n has access to an oracle ...

**6**

votes

**0**answers

254 views

### Is integer GCD in NC?

Wikipedia, in the page http://en.wikipedia.org/wiki/Greatest_common_divisor#Complexity mentions integer GCD is in NC by citing http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/Sedjelmaci07.pdf
...

**1**

vote

**2**answers

115 views

### complexity of finding optimal matchings of given fixed size

It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...

**6**

votes

**1**answer

245 views

### Do sparse DAGs can have large min-cuts?

For a graph $G$, let $e(G)$ denote the number of its edges, and $c_k(G)$ the smallest number
of edges that must be removed in order to destroy all paths of length $\geq k+1$.
Note that $c_1(G)\geq ...

**16**

votes

**1**answer

606 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...

**1**

vote

**0**answers

130 views

### NP problem implications [closed]

Hi, i would like to have some clarification on NP-completeness.
In particular I'm reading an article where they show:
1) Partitioning the edges of a graph into connected component of 3 edges (3-path ...

**4**

votes

**1**answer

228 views

### If a graph invariant is NP-Hard, is its “deck ratio” NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define ...

**8**

votes

**1**answer

369 views

### computational complexity

I am interested into computational complexity of decision problem: Does a given 2-dimensional simplicial complex contain (any)triangulation of 2-sphere? This problem trivially lies into NP, because ...

**10**

votes

**6**answers

1k views

### Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity.
The wikipedia article on constructive proof begins, "a constructive ...

**5**

votes

**0**answers

110 views

### 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} ...

**6**

votes

**1**answer

258 views

### How do you compute the primes of bad reduction?

Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some ...

**4**

votes

**1**answer

437 views

### Exact arithmetic for real algebraic numbers

There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...

**12**

votes

**6**answers

1k views

### SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...

**1**

vote

**1**answer

330 views

### Non-uniform complexity of the halting problem

This question is approximately cross-posted from Theoretical Computer Science Stack Exchange: http://cstheory.stackexchange.com/questions/14445/complexity-of-the-halting-problem
What can be said ...

**2**

votes

**1**answer

258 views

### If NP=EXPTIME, does every DTM have a succinct “execution proof”?

Let HALTS-IN-N be the canonical $EXPTIME$-complete language {<$M$,$n$> | the deterministic Turing machine (DTM) encoded by $M$ halts in $n$ or fewer steps, with $n$ encoded in binary}. Since ...

**3**

votes

**2**answers

99 views

### How would one characterize a PR-complete language?

The complexity class $PR$ is the set of all formal languages that can be decided by a primitive recursive function. Is there any language $l$ known to be complete for this class, i.e., for every ...

**3**

votes

**1**answer

146 views

### fast approximate k-nearest neighbors in high dimensions?

Hi, I've been scanning the literature trying to find an adequate approximate k-neighbour for my outlandish data set, but I remain stymied. Perhaps someone can help?
The dataset is huge, both in ...

**8**

votes

**3**answers

822 views

### shallow question: Why a 300 digit number is associated with “any NP-hard problem”?

I was reading this article (http://www.ams.org/notices/200203/fea-knuth.pdf) the other day and noticed Donald Knuth said something nontrivial: Theoretically we can compute a very large number of ...

**2**

votes

**0**answers

110 views

### supersingular curve detector

Suppose I give you a prime $p$ and ask for a non-CM supersingular elliptic curve over $\mathbb{F}_p.$ Can this be done in polynomial time (so, polynomial in $\log p$)?

**0**

votes

**1**answer

250 views

### Is unconstrained integer convex optimization problem NP-hard?

Does anyone know a reference to the answer if unconstrained integer convex optimization problem (i.e. $\min_{x\in \mathbb{Z}^N} F(x)$, $F$ is convex and $N$ is NOT fixed) is NP-hard?
Thank you in ...

**7**

votes

**3**answers

263 views

### Can we efficiently compute a third Nash Equilibrium, given two?

A finite, two-player, nondegenerate, symmetric game is defined by a nondegenerate $n \times n$ payoff matrix $A$. If player 1 plays strategy $i$ and player 2 plays strategy $j$, then player 1's ...

**2**

votes

**1**answer

212 views

### Typical dimension of partial derivatives

Let $V$ be the space of all homogenous polynomials over $\mathbb{C}$ in $n$ variables of degree $d$.
Let $l,k$ be two integers and $f\in V$.
Let $\partial^{=k}(f)$ be the space of all partial ...

**2**

votes

**1**answer

253 views

### Complexity of establishing finite groups (non)-isomorphism ?

Question Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N ...

**3**

votes

**1**answer

145 views

### Functions That Can Be Computed Faster Simultaneously Than Expected

The following is an elementary question about circuit complexity. It is different from the kind of thing I have seen discussed, so I would be interested in any work that has been done on this kind of ...

**7**

votes

**2**answers

835 views

### How to compute all irreducible representations of a finite group ? (how GAP is doing this?)

Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
Question: How ...

**2**

votes

**2**answers

262 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**1**

vote

**2**answers

134 views

### Is number of quasi-kernels NP-hard?

A quasi-kernel in a directed graph D is an independent subset of vertices $S$ so that for every $v \in V(D)-S$ either $v->s$ for some $s \in S$ or $v->w->s$ for some $w \in V(D)-S, s \in S$.
...

**0**

votes

**1**answer

95 views

### Provided a list of sets, $L$, computing an array where each entry $q_i \in Q$ is the family of sets in $L$ that have intersection $k$ with $l_i \in L$

I have a set of $(l_1, ..., l_N) \in L$ smaller sets, each with $(r_1, ..., r_M) \in R$ integer elements. I would like create an ordered array of $(q_1, ..., q_N) \in Q$ sets s.t.:
(1) Each $q_i \in ...

**1**

vote

**1**answer

195 views

### Redundancy and Structure of computational problems

It is widely believed that some computational problems such as graph isomorphism can not be NP-complete because it does not possess enough structure or redundancy to be computationally hard (NP-hard). ...