**7**

votes

**1**answer

466 views

### Doob Martingale: Where is the catch?

I am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support.
I am attempting to use the method of bounded ...

**2**

votes

**1**answer

425 views

### Polygamous stable marriage/ assignment problem

I'm not sure under which 'algorithm' it falls under, but here is the problem:
I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...

**1**

vote

**2**answers

51 views

### Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...

**4**

votes

**0**answers

65 views

### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...

**-1**

votes

**0**answers

30 views

### algorithm to split one real number to 3 variables [closed]

I asked this question on stackoverflow already but maybe it is more math related.
I try to store integer (real numbers) values into pixel data. The only way my api can store pixel data are RGB ...

**9**

votes

**1**answer

173 views

### Exact determinant of a circulant matrix

The wikipedia gives us a formula for the determinant of a circulant matrix. That is:
$$\mathrm{det}(C)
= \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1})= ...

**4**

votes

**1**answer

169 views

### Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields?
For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...

**3**

votes

**2**answers

560 views

### Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...

**2**

votes

**0**answers

58 views

### Detecting Negative Cycles in Undirected Graphs

I recently faced the problem of quickly detecting negative cycles in undirected, weighted graphs. Resorting to the Bellman-Ford Algorithm, as commonly suggested, turned out to be very inefficient and ...

**11**

votes

**2**answers

3k views

### All-pairs shortest paths in trees?

This is a reference request, since I'm sure what follows isn't new, but I can't seem to find it.
Suppose that we have a finite tree $T$ with non-negative weights on the edges. Naively, computing the ...

**2**

votes

**0**answers

45 views

### Is there a name for this variant of the MST and the TSP?

Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...

**0**

votes

**1**answer

237 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 ...

**4**

votes

**1**answer

127 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 ...

**3**

votes

**0**answers

71 views

### Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this?
...

**10**

votes

**3**answers

488 views

### Algorithm for detecting ribbon or slice links?

A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to ...

**14**

votes

**2**answers

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 ...

**16**

votes

**2**answers

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 ...

**2**

votes

**1**answer

100 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 ...

**11**

votes

**1**answer

627 views

### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned:
'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...

**2**

votes

**0**answers

9 views

### Complexity of OBDD isomorphism (representing same function after permutation of variables)?

According to wikipedia Ordered Binary Decision Diagarams (OBDD) are a data structure that is used to represent a Boolean function.
OBDD is a DAG with two sinks $0,1$.
The size of the BDD is number ...

**2**

votes

**0**answers

48 views

### Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...

**3**

votes

**0**answers

71 views

### Algorithm for finding the fewest cartesian products to partition a set of points

The question Algorithm for finding the fewest rectangles to cover a set of rectangles was already answered here and a similar question was answered here. Those questions were about regular geometric ...

**32**

votes

**19**answers

4k views

### What is the easiest randomized algorithm to motivate to the layperson?

When trying to explain complexity theory to laypeople, I often mention randomized algorithms but seemingly lack good examples to motivate their usage. I often want to mention primality testing but ...

**8**

votes

**8**answers

12k views

### Pseudo-random number generation algorithms

What algorithms are used in modern and good-quality random number generators?

**1**

vote

**3**answers

258 views

### Finding integer representation as difference of two triangular numbers

Since $n = \frac{n(n+1)}{2}-\frac{n(n-1)}{2}$, every natural number can be represented as the difference of two triangular numbers:
$ n = \frac{a(a+1)}{2}-\frac{b(b-1)}{2}$. Finding such a ...

**5**

votes

**3**answers

4k views

### What is a good algorithm to measure similarity between two dynamic graphs?

I am using graphs to represent structure present in a scene. The vertices represent the objects in the scene and the edges represent the relationship between two nodes(touching, overlapping, none). ...

**1**

vote

**1**answer

35 views

### Test Instances for Perfect Matchings in Graphs

Are there any graphs with a known set of perfect matchings and other predefined properties, such as vertex connectivity, which can be used for testing the implementation of matching algorithms?
...

**4**

votes

**2**answers

1k views

### Factoring and solving trinomials

Has the problem of factoring (over the rationals) the general trinomial $ax^n+bx^k+c$ with $a,b,c\in\mathbb{Z}$, $n,k\in\mathbb{N}, n>k>1$ been solved? By solved I mean a classification theorem ...

**3**

votes

**0**answers

57 views

### What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when ...

**3**

votes

**2**answers

148 views

### Bellman-Ford for Matching Problems?

I am looking for a simple way of calculating minimum-weight perfect matchings in complete graphs with an even number of vertices.
I know that there are implementations that are based on Edmond's ...

**1**

vote

**1**answer

73 views

### What is the relation between size of maximum clique and branchwidth?

Let $bw(G)$ be the branchwidth of graph $G$ and $\omega(G)$ be the size of maximum clique in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
Intuition: Assume (in reverse of ...

**0**

votes

**1**answer

76 views

### The cost function in the Weighted Bipartite Matching Problem (a.k.a the Assignment Problem)

In the definition of this problem, the weight/cost function generally takes value in $\mathbb{Z}$ (or sometimes $\mathbb{Q}$).
This is what I observed from some books (e.g. "Combinatorial ...

**8**

votes

**1**answer

448 views

### Some Questions on the Collatz conjecture

The set of all positive whole numbers is denoted by $\mathbb{N}_+$.
Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto
\begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ ...

**1**

vote

**0**answers

55 views

### Does Coppersmith's method always finds non-trivial factor of integers of the form $n=a(2^k b+1)$ assuming $1 < a<2^k b +1$ and $b < n^{1/4-0.05}$?

Got an argument and numeric evidence that pari's implementation
of Coppersmith's method finds non trivial factor of integers
of certain form under some assumptions very efficiently.
Three $5000$ bit ...

**6**

votes

**1**answer

266 views

### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...

**0**

votes

**0**answers

65 views

### Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...

**6**

votes

**1**answer

91 views

### Algorithm that solves every Mixed Integer Linear Program (to optimality)?

Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically?
I know that you usually ...

**6**

votes

**0**answers

71 views

### Correlation of Class Functions

Let $G$ be a finite group, and let $f_1,f_2$ be two real-valued class functions of $G$. Assume that multiplying elements of $G$ takes $O(1)$-time.
Let $s:G\to \mathbb{R}$ be defined by ...

**0**

votes

**0**answers

19 views

### Monotonicity per dimension of multivariate scattered data

For my thesis, I am working on interpolation using the RBF method (Radial Basis Functions). Before interpolating, I want some a priori insight into the data, for example check in which dimensions it ...

**4**

votes

**0**answers

169 views

### Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...

**4**

votes

**2**answers

141 views

### Extraction of Coefficients in the Exponential Function of a Series

Question: Let $f(x) \in x\mathbb{C}[[x]]$. What is the (asymptotically) fastest algorithm for calculating the coefficient of $x^n$ in $e^{f(x)}$?
Naive Solution 1: Using fast polynomial ...

**3**

votes

**1**answer

241 views

### Reducible polynomials

Say one only seeks to identify whether a given polynomial over $\mathbb{Z}[x]$ is reducible, then what are the best ways known to solve this?
$(1)$ If the polynomial is reducible, the algorithm ...

**3**

votes

**1**answer

150 views

### Algorithm to count the number of perfect matchings in non planar graph

I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing ...

**2**

votes

**2**answers

334 views

### Minimally intersecting subsets of fixed size

The question I have, is how to generate the following collection of subsets:
Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...

**-1**

votes

**1**answer

82 views

### Is there an algorithm to find a linear dependence between points on elliptic curves?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...

**11**

votes

**2**answers

420 views

### Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)

Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.
Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix ...

**2**

votes

**1**answer

78 views

### Seeking for abelian subalgebra of fixed dimension in finite Lie algebra

The problem is: I want to know if there is abelian subalgebra of dimension $k$ in Lie algebra of dimension $n$. My Lie algebra is given by its structure constant table. There are some algorithms ...

**1**

vote

**2**answers

56 views

### Maximum subgraph edge distance greater than given number

I have a weighted graph G with approximately 75000 nodes. I would like to find subgraph G' induced on a subset of nodes, such that all edge weights in G' are greater than a given constant C and the ...

**0**

votes

**2**answers

295 views

### Enumerating m-tuples of Integers Subject to Implication Constraints [closed]

How do I enumerate all $m$-tuples of positive integers $(a_1,...,a_m)$ subject to the following constraints?
For each $i$ in $\{ 1,\ldots,m \}$, there is a number $n_i \geq 0$ such that $a_i \leq ...

**11**

votes

**1**answer

285 views

### Travelling salesman: can the furthest-neighbour algorithm beat the nearest-neighbour?

This is a problem that has bugged me for quite some time, and I have not been able to find any documentation about it online. It is well known that the NN algorithm can yield the worst possible route ...