Informally, an algorithm is a set of explicit instructions used to solve a problem (e.g. Euclid's algorithm for computing the greatest common divisor of two integers). For more specific questions on algorithms, this tag may be used in conjunction with the approximation-algorithms, ...

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7
votes
1answer
117 views

Determine if an $n$-dimensional mesh of simplices is a non-manifold

In an $n$-dimensional space I have a set of simplices where each simplex consists of facets. Some of the simplices are 'connected' by sharing facets. Each facet is made up on edges, each consisting ...
21
votes
2answers
812 views

Groups where word problem is solvable, but not quickly?

Are there finitely generated groups whose word problem is solvable, but not quickly? It would be great to have specific examples, but existence results would also be helpful. All of the groups that ...
5
votes
0answers
92 views

Diameter of the modified bubble-sort graph

The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
0
votes
0answers
171 views

Best measure for curve similarity

I would like to measure similarity between two curves represented by two arrays of points. The similarity measure should not depend on the size of these shapes. Two similar shapes but have different ...
5
votes
4answers
402 views

Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before: Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...
2
votes
0answers
280 views

Partitioning graph for Graph Isomorphism [closed]

Motivation: I am studying the graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases . Construction: $G$ is an $r$ regular graph, $k$ connected (not a ...
2
votes
1answer
60 views

Maximal opening angle of a polygon from a point [closed]

I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point. One ...
3
votes
1answer
89 views

Equality of Euclidean numbers / constructible numbers

Euclidean numbers are those real number that can be constructed from the natural numbers by a finite chain of +,-,*,/ and $\sqrt{}$. They are also called Constructible Numbers. I am now interested in ...
5
votes
1answer
113 views

Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity? Especially modeling ...
2
votes
0answers
203 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
10
votes
6answers
1k views

Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...
10
votes
1answer
830 views

Travelling Salesman Problem: Can the nearest neighbor algorithm be $n$ times longer than the optimal solution?

This is inspired by a recent question. Given a positive integer $n\in\mathbb{N}$, is there a setting of finitely many points and a designated "starting point" $s$ in $\mathbb{R}^2$ such that the ...
0
votes
2answers
132 views

Generate all non-isomorphic partitions $\pi = \{ \{1, …, n-1\}, \{n\} \}$ for all graphs of order $n$

Let $G$ be any connected, undirected, and unweighted graph of order $n$. Let $\pi = \{ \{ 1, ..., n-1 \}, \{ n \} \}$ be partitioning of $G$ such that always $n-1$ vertices are in the first cluster ...
4
votes
0answers
44 views

Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected. You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...
1
vote
1answer
186 views

Finding the “minimum norm” of points in projective space above a prime field

I am working on doing explicit computations finding class groups of quaternions over $\mathbb{Q}$, and the following question (with $n=3$) was a clear choking point: Given a point $P = [x_0, \dots, ...
5
votes
4answers
211 views

From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix

Suppose I am given a symmetric matrix $G_{ij}$ with $G_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)_{ii}=2$ and $(S^T G S)_{ij} \leq 0$ for $i \neq j$? Is there a ...
-4
votes
1answer
174 views

Without the use of a calculator, how to calculate the logarithm of 2 and 3 in base 10 [closed]

Without the use of calculator how to calculate $\log_{10} ~2$, $\log_{10} ~3$?
1
vote
1answer
175 views

Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?

I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ...
8
votes
1answer
146 views

Computing a transversal of a subgroup $H$ of $G$ in expected $O(|G : H|^2 \log |G : H| + |H|)$ time

I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup $H$ of $G$ is a single ...
0
votes
0answers
83 views

Finding a “special” non singular submatrix

Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ ...
7
votes
2answers
167 views

Isomorphism problem on the class of induced subgraphs of a hypercube

A problem that I am currently studying translates to the problem of deciding whether two induced subgraphs of the hypercube $Q_k$ are isomorphic. Now it feels to me that this class of graphs is "too ...
5
votes
0answers
167 views

Is there a reasoned derivation of the coherence conditions for symmetric rig categories?

I know what the coherence conditions are, I can look them up in M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72. In theory, ...
1
vote
1answer
138 views

Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...
8
votes
0answers
155 views

Complete list of exceptions and efficient algorithm for Waring's problem

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...
2
votes
0answers
73 views

A basic minimization problem over finite fields

Let $p$ be a prime, and suppose we are given $n$ values mod $p$: $a_1,...,a_n\in Z_p$. Is there a fast algorithm for finding $\alpha\in Z_p$ which minimizes the value $\max_i (\alpha\cdot a_i$ mod ...
5
votes
1answer
367 views

How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field. Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra. Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take ...
4
votes
2answers
422 views

Breaking a rectangle into smaller rectangles with small diagonals

Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
3
votes
2answers
197 views

Paper of Denis Simon on quadratic equations in dimensions 4, 5?

In several places I have come across references to a 2005-6 preprint of Denis Simon entitled Quadratic equations in dimensions 4, 5, and more This paper gives fast algorithms to find isotropic ...
6
votes
1answer
1k views

Constructing the oracle for Grover's algorithm

For a final project in my class, I decided to try to simulate a quantum computer and implement Grover's algorithm. I followed this excellently written blog post by Craig Gidney, and was successful in ...
10
votes
4answers
543 views

Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...
4
votes
0answers
174 views

What is the complexity of intersecting two matrix algebras over a finite field?

The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi. Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$. For a ...
3
votes
1answer
378 views

Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
12
votes
1answer
592 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
4
votes
0answers
195 views

Minimum number of real multiplications to multiply two quaternions [closed]

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
5
votes
2answers
212 views

Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community. Setup: Let ...
4
votes
0answers
152 views

What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
0
votes
1answer
112 views

Is there a closed form for tan(q*pi) with q rational? [closed]

I'm looking for a closed-form expression for tan (q*pi) for q rational, or an algorithm that generates one, or some other means of compactly describing the closed-form without referencing an infinite ...
1
vote
2answers
275 views

Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$

I want to fast decompose polynomial over ring of integers (original polynomial has integer coefficients and all of factors have integer coefficients) and also over ring of integers modulo prime ...
4
votes
1answer
73 views

Smallest sum $S \geq k > 0$ using one element from each of several sets of nonnegative integers

Does there exist a way to efficiently solve the following problem? Given some constant $k$ and several sets of non-negative integers: ...
3
votes
1answer
125 views

What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem?

I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered. So, ...
0
votes
0answers
34 views

Complexity of optimizing a bi-objective function with integer constraints

I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...
2
votes
0answers
100 views

Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...
1
vote
2answers
234 views

Curves similarity metric [closed]

I am working on an optical character recognition algorithm that takes vector data (i.e. polylines) as input rather than raster picture. E.g., we have N polyline samples, and when certain polyline is ...
13
votes
1answer
404 views

Computer software for periods

Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...
1
vote
1answer
60 views

Hamiltonian Path through $n$-bit strings with maximum number of $0\mapsto 1$ transitions

Let $G_n$ be the complete graph whose vertices are the $2^n$ $n$-bit strings. Let $H_n$ denote the Hamiltonian path through $G_n$ that uses the maximum number of edges that correspond to a single bit ...
2
votes
0answers
66 views

Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M

this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :) Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...
1
vote
0answers
72 views

Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations $$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$ where $\alpha=(\alpha_1,...,\alpha_s)$ and ...
6
votes
4answers
437 views

Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers ...
2
votes
0answers
203 views

Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? [closed]

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). It seems to me that much ...
2
votes
1answer
156 views

Is there any algorithm can find local minima of nonconvex objective function in guaranteed polynomial time?

More precisely, The setting could be formulated as, $min. F_{\lambda}(p)$ over permutation matrices $P$ Here $F_{\lambda}(p)$=$\lambda *F_{0}(p)+(1-\lambda)F_{1}(p)$ where both $F_{0}(p)$ and ...