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13
votes
0answers
135 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 ...
0
votes
0answers
46 views

Computing a Levenshtein edit distance between two strings when a particular set of string edits is forbidden

Let's imagine we'd like to compute a Levenshtein edit distance between two strings, $S_1$ of length $n_1$ and $S_2$ or length $n_2$, consisting of characters over a finite alphabet $R$ of size $q$ ...
3
votes
1answer
157 views

Generalization of notion of convexity

I am searching for the correct term for the following, if it exists. A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius ...
1
vote
0answers
41 views

Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...
3
votes
1answer
79 views

A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not. The new problem is this. There is a tourist who has a having the following ...
2
votes
0answers
87 views

coin reversal puzzle with one hand and two stacks

Suppose that you have N labeled coins pinched in one stack in your fingertips (your palm is above your fingers and your palm is facing down, so that you can drop as many coins as needed from the ...
1
vote
0answers
88 views

Combinatorial design for minimization problem over binary strings

Suppose the cost of a binary string $B$ of length $k$ is the number of $1$s that occur before the last $0$. For example, $1110$ has cost 3 while $0111$ has cost 0. Now suppose you can choose $k$ ...
0
votes
0answers
48 views

Find minimal set of progressions which intersections, unions or negations covers given set

Given an integer $N$ and a set of integers in $[1; N]$. Find a minimal set of integer arithmetical progressions such as given set can be covered using operations $A \cap B$, $A \cup B$ and $\overline ...
-2
votes
0answers
93 views

Given a set of generators of a group G, is there a method to find a presentation for G using those generators? [migrated]

Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want). Suppose I have a small list of generators ...
1
vote
1answer
65 views

Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language: $L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$. ($G$ ...
6
votes
2answers
142 views

get a point in polygon (maximize the distance from borders)

I have several 2D polygons represented by lists of xy-coordinates of their vertices. It is needed to get several points inside the polygon so that they lie possibly far from the polygon's borders ...
2
votes
2answers
158 views

Integers up to n having an even number of trailing zeros in their factorial

It is well-known that the number of trailing zeros in the factorial $k!$ is given by the nice function $$ z(k) := \sum_{i \ge 1} \left\lfloor \frac{k}{5^i} \right\rfloor. $$ Now assume that we want ...
0
votes
0answers
69 views

Minimize the length of two disjoint segments in the string with given property

You are given a string s of size n, consisting of characters A and B only. You have to find minimum sum of size of the two disjoint segments of the string s such that number of A's in them are >= z. ...
3
votes
1answer
49 views

Random weighted selection without replacement

I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$. Selection procedure ...
4
votes
4answers
311 views

Determine if a graph has a large clique

This question is quite specific and practical. I hope it is still relevant for MO and will not be removed. I have a collection $\mathcal{C}$ of graphs having from 5000-6000 vertices and edge density ...
2
votes
0answers
48 views

Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...
2
votes
0answers
23 views

largest size for a randomness extractor

I am not so expert in theoretical computer science, so sorry if the question is trivial, i just could not find it in literature. Suppose we have a source $X$ with min-entropy $\ell$, the randomness ...
0
votes
0answers
33 views

Multi-objective set-cover optimisation problem

I'm looking for an algorithm to solve the following multi-objective set-cover problem. We start with a 'universe' (set) of items $\mathcal{U}$, along with a partitioning $P = \{p_0,\ldots,p_m\}$ ...
1
vote
0answers
94 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I want ...
3
votes
0answers
126 views

What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
3
votes
1answer
147 views

How to perform Importance Sampling with Prior Information

Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as ...
3
votes
0answers
88 views

Generating random weak k-bounded reverse plane partitions

Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that ...
0
votes
0answers
63 views

approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...
4
votes
1answer
105 views

Is there an efficient algorithm to check whether a matrix is symmetrizable using only permutation matrix?

Is there any known efficient algorithm (something that works better than brute-force algorithm) to check that given a $(0,1)$-$d \times d$ matrix $A$, is there a permutation matrix $P$ of the same ...
4
votes
2answers
134 views

Details of generation programs supplied with nauty

The program nauty comes with gtools which contains, among others, several generation programs like geng, genbg, ... I was ...
1
vote
4answers
165 views

Counting simple 4-cycles in an undirected graph [closed]

I'm looking for an algorithm which just counts the number of simple and distinct 4-cycles in an undirected graph labelled with integer keys. I don't need it to be optimal because I only have to use it ...
3
votes
0answers
113 views

Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task: Let $\mathbb{F}$ be a finite field. Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let $G=\langle ...
2
votes
1answer
329 views

Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq ...
6
votes
1answer
228 views

Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product $$ p=b_1 b_2 \cdots b_n$$ where each $b_i\in A$. Clearly $n-1$ multiplications suffice to compute $p$; ...
2
votes
2answers
175 views

Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...
3
votes
2answers
178 views

Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances. The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
2
votes
1answer
198 views

efficient arithmetic with (short) Conway games?

We consider "games" in the sense of ONAG. Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations ...
1
vote
0answers
51 views

Finding special vectors generated by a matrix

Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix. Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
1
vote
0answers
41 views

Covering a set of points by bounded geometric object/objects

1) Let $S$ be a set of $n$ points in $R^d$. Now, given a bounded geometric object $G$, the problem is to check whether $S$ can be contained in $G$. 2) Also, in general setting, the problem is to ...
6
votes
0answers
135 views

Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...
2
votes
1answer
111 views

Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...
2
votes
1answer
124 views

Are there any good techniques for calculating Hausdorff measure?

I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...
9
votes
1answer
393 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 ...
1
vote
1answer
195 views

Does this algorithm terminate in all scenarios?

Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x_1, \dots, x_n\}$ and $B = \{x_{n+1}, \dots, x_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in ...
8
votes
1answer
457 views

How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
-2
votes
1answer
213 views

Polygon Problem [closed]

There are $N$ regions which are numbered from $1$ to $N$. Each region is represented by a single simple polygon on the 2D plane. Simple polygon means the boundary of the polygon does not cross itself. ...
7
votes
2answers
204 views

How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...
0
votes
1answer
92 views

Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...
0
votes
0answers
65 views

Minimum cut in near complete graph

I have a problem with an algorithm that i don't quite understand. The mathematics are not the problem, but I don't understand how it can work in practice. The problem is in the paper ...
1
vote
1answer
152 views

Set of distributions that minimize KL divergence,

Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence, is there a ...
1
vote
0answers
78 views

method to construct platonic solids from the circumscribed sphere [closed]

I am trying to find a geometrical method of building the platonic solids starting from the circumscribed sphere. I do have a class of generative 3d modeling where we create 3d object programmatically ...
6
votes
4answers
472 views

How long does it take to compute a class number?

I was wondering if there are any known (upper and lower) bounds for the complexity of computing the class-number of a finite extension of the rationals. (A general bound should be in function of the ...
20
votes
4answers
1k views

Securing privacy of “who communicates with whom” under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data ...
1
vote
2answers
106 views

Draws from multiple non-disjoint urns

Let $C = \{1,...,n\}$ be a set of $n$ colors. Let $S_1,...,S_k$ be non-empty subsets of $C$, that is, $S_i \subseteq C$ for all $i \in \{1,...,k\}$. It is helpful to think of the $S_i$ as urns with ...
3
votes
2answers
203 views

Complexity of a problem remotely related to the discrete logarithm $A=x g^x$

Let $x,g \in \mathbb{F}_p^\ast$. Given $g$ and either $$ A = x g^ x$$ or $$ A = x g^{x^2-1}$$ find $x$. What is the complexity of solving this? Is there a reduction to the discrete ...