computational complexity theory; complexity classes, such as P, NP, PSPACE, and so on; resource-limited computation; NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models such as automata, circuits; regular languages; ...

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11
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308 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
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0answers
92 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
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2answers
224 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
4answers
798 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
1answer
183 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
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0answers
72 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
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1answer
154 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
3answers
1k 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
1answer
107 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
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0answers
122 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
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2answers
732 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
2answers
537 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
0answers
57 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
1answer
257 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 ...
17
votes
0answers
430 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
1answer
220 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
1answer
202 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
2answers
128 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
0answers
245 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
2answers
111 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
1answer
236 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
1answer
571 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
0answers
129 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
1answer
226 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
1answer
362 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 ...
9
votes
6answers
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
0answers
107 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
1answer
243 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
1answer
391 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
6answers
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
1answer
301 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
1answer
236 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
2answers
96 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
1answer
144 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 ...
7
votes
3answers
801 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
0answers
109 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
1answer
226 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
3answers
241 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
1answer
211 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
1answer
225 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
1answer
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
2answers
716 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
2answers
250 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
2answers
131 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
1answer
93 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
1answer
189 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). ...
1
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0answers
90 views

Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?

Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard? Equally interesting would be to learn about such problems with a ...
2
votes
1answer
203 views

Does Quadratic Programming get easier when it's described by a diagonal matrix?

Generally, Quadratic Programming solves the problem $$\text{Given }Q, c, A, b,\text{ choose }x \text{ to maximize } x^TQx + c^Tx \text{ subject to } Ax \le b$$ In this form, Quadratic Programming is ...
2
votes
0answers
176 views

A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete. I provide two simple definitions and one example at first. Definition 1. Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...
2
votes
1answer
111 views

Any result or conjecture of computaional complexity of formal languange with rational generating function?

As we know that context-free language is in P,any result or conjecture of computaional complexity of formal languange with rational generating function?And more,any result or conjecture of ...