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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21
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2answers
3k views

Distinct numbers in multiplication table

Consider multiplication table for numbers $1,2,\cdots, n$. How many different numbers are there? That is how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there? I'm interested ...
23
votes
10answers
11k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
5
votes
1answer
513 views

Difference Sets

Suppose $$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$ We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$ Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...
16
votes
2answers
1k views

Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...
42
votes
4answers
11k views

How hard is it to compute the number of prime factors of a given integer?

I asked a related question on this mathoverflow thread. That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered. ...
27
votes
1answer
469 views

Decidability of equality of expressions built using 1,+,-,*,/,^

Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...
14
votes
3answers
2k views

When are (finite) simplicial complexes (smooth) manifolds?

Hi, is there an algorithm that determines if a given simplicial complex is a.) a manifold b.) a smooth manifold c.) homotopy equivalent to a manifold d.) a real algebraic variety ?
12
votes
2answers
463 views

Checking whether given binary operation is a group operation

Given a binary function $f: [1..n] \times [1..n] \to [1..n]$ how to check that this operation is a group operation on $[1..n]$? It's obvious that this can be done in $O(n^3)$ time just by checking ...
9
votes
1answer
386 views

Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction? The answer does ...
6
votes
1answer
900 views

Algorithm for embedding a graph with metric constraints

Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an ...
3
votes
3answers
3k views

Algorithm for detecting prime powers

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote: "This scheme will thus work as ...
4
votes
1answer
292 views

an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$. I mean there is polynomial reduction $F$ such that for every boolean ...
7
votes
2answers
454 views

What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath (Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...
6
votes
2answers
315 views

Minimum number of unit fractions to sum up a given positive rational

For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum_{s\in S} s=\frac{p}{q}$, see this article by Paul Erdös and Sherman Stein ...
27
votes
5answers
7k views

Computing the Galois group of a polynomial

Does there exist an algorithm which computes the Galois group of a polynomial $p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of ...
27
votes
9answers
12k views

Fast Matrix Multiplication

Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in ...
31
votes
2answers
2k views

Does anyone want a pretty Maass form?

A few months ago, I was curious about some properties of Maass cusp forms, of nonabelian arithmetic origin. As a result, I went through a somewhat predictable process of finding a totally real $A_4$ ...
37
votes
1answer
2k views

improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
19
votes
4answers
2k views

does the “convolution theorem” apply to weaker algebraic structures?

The Convolution Theorem is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse ...
16
votes
1answer
3k views

Evidence for integer factorization is in $P$

Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter ...
11
votes
4answers
2k views

Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ? (Main case - complex numbers, comments on other cases are also welcome. "Given" ...
21
votes
0answers
419 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
17
votes
4answers
2k views

Complexity of testing integer square-freeness

How fast can an algorithm tell if an integer is square-free? I am interested in both deterministic and randomized algorithms. I also care about both unconditional results and ones conditional on GRH ...
10
votes
6answers
1k views

Where on the internet I can find database of graphs?

I am studying graph algorithms. I need database of graphs on which I can test my algorithms. Where can I find reliable database of graphs of all kinds? Thanks!
9
votes
2answers
477 views

When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?

To ask this question in a (hopefully) more direct way: Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ ...
13
votes
2answers
1k views

Detecting almost-primes quickly

There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product ...
12
votes
8answers
6k views

Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...
6
votes
2answers
625 views

Differential ideal membership problem

We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case? To be ...
19
votes
3answers
1k views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in ...
14
votes
6answers
6k views

Good algorithm for finding the diameter of a (sparse) graph?

My question on Stack Overflow was recently tagged "math". Despite a bounty, it never received a satisfactory answer, so I thought I would ask it here: I have a large, connected, sparse graph in ...
12
votes
1answer
542 views

Stable matchings when switches have costs

The Gale-Shapley algorithm finds a stable matching in the complete bipartite graph, for any preference matrix. It's also well-known that stable matchings don't always exist in the complete graph ...
6
votes
1answer
436 views

Is equality of terms for “real” numbers with roots, logarithm, exponential, sin, cos, and other trigonometric operations decidable with a Turing-machine?

If yes, how? Also, I know you can't do it for arbitrary statements about real numbers, but that's not what I'm asking, and by "real" numbers, I mean the numbers constructible from 1, -, /, and the ...
5
votes
1answer
187 views

Algorithm for Low Discrepancy Sequence of Hamilton Cycles in Complete Graphs

Are there deterministic algorithms that generate a sequence of Hamilton Tours that is superior to a sequence of randomly chosen tours, when applied to the TSP (by applying to the TSP, I understand ...
4
votes
2answers
257 views

What's the current state of one-rule semi-Thue system termination problem?

What's the current state of one-rule semi-Thue system termination problem? Search produces a lot of references, but it's hard to find out if decidability of this problem has been proven or not.
4
votes
2answers
1k views

How to compute the rook polynomial of a Ferrers board?

Given a Ferrers board of shape $(b_1,\ldots,b_m)$, we define $r_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's ...
3
votes
0answers
282 views

What is God's number for the WrapSlide puzzle?

WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
29
votes
3answers
2k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
16
votes
2answers
4k views

Generalization of the shakehands/condom puzzle?

The classic handshake puzzle goes something like this: "Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?" Its common ...
10
votes
1answer
293 views

Network flows with capacities on pairs of edges

Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$. Now add edge-pair ...
10
votes
3answers
473 views

Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
7
votes
1answer
1k views

How much does a quantum oracle to find a needle in a haystack really cost?

Among the basic algorithms of quantum computations Lov Grover's result on quantum search stands out, both in regards to its intrinsic interest, and for its undisputable elegance. Grover's algorithm ...
7
votes
4answers
2k views

Stirling Number of first kind : Implementation

Hi everybody, Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula $$ x(x-1)\cdots (x-n+1) = \sum_{k=0}^n s(n,k)x^k. $$ Otherwise, what is ...
6
votes
2answers
2k views

Can a number be factored quickly, given the sum of its prime factors?

This is perhaps most naturally phrased as a promise problem. Given numbers $n$ and $s$, where $s$ is the sum of the prime factors of $n$ (distinct or with multiplicity; I imagine both variants will ...
4
votes
0answers
195 views

Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
1
vote
2answers
157 views

Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity. ...
9
votes
3answers
1k views

Mertens' function in time $O(\sqrt x)$

This MathOverflow question seems to indicate that the state of the art in computing $$ M(x)=\sum_{n\le x}\mu(n) $$ takes time $\Theta(n^{2/3}(\log\log n)^{1/3}),$ which matches my understanding. ...
7
votes
1answer
421 views

Building a polyhedron from areas of its faces

Is there a known algorithm which, given a finite multiset (unordered list) of integers $A$, returns a yes/no answer for ...
6
votes
2answers
325 views

Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.

First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
5
votes
0answers
183 views

Differential operators that preserve real-rootedness

Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of ...
2
votes
1answer
105 views

Does anyone have the correct link to treewidth.com?

Several posts (and this) on StackExchange sites like MO have some link-rot. For example, I've been looking into tree decomposition and keep coming across references to treewidth.com, but the link ...