All Questions
Tagged with ds.algorithmsapproximation-algorithms
169 questions
0votes
0answers
102views
Understanding a remark on O(log d) ratio for Online Set Cover
In the paper "The Online Set Cover Problem" by Alon, Awerbuch, Azar, Buchbinder and Naor, they study an online version of the set cover problem in which elements arrive one by one and ...
- 649
0votes
0answers
49views
Min cost perfect matching, but with conflicting edge pairs
Consider the following variant of min-weight perfect matching. We are given a graph $G = (V,E)$ with non-negative weights on the edges. We are also given a collection of pairs of edges $P \subseteq E \...
- 649
4votes
1answer
157views
Min-cost perfect matching, but must pick exactly k special edges. Is it NP-hard?
I'd like to know if the following generalization of min-cost perfect matching is NP-hard. As usual, we are given a graph $G = (V,E)$ with costs on edges $c: E \to \mathbb{R}_{\geq 0}$. In addition, ...
- 649
1vote
0answers
172views
Is finding the best permutation an NP-Complete problem?
We have a matrix $M$ of size $n$ by $n$ where $M[i][j] \ge 0$ and $M[i][i] = 0$. We want to create a permutation of integers $[1,\dots,N]$, like $\langle P_1, P_2,\dots,P_n \rangle$, such that $$ \...
- 554
3votes
1answer
136views
How well can shortest common supersequence over small alphabet size be approximated?
Given a list $L$ of sequences of the first $n+1$ natural numbers, how well can we approximate the shortest common supersequence of all sequences in $L$? The paper here shows that if $n$ is not ...
- 219
2votes
0answers
87views
References for algorithms to compute approximating polytopes for arbitrary convex sets
There is a vast theoretical literature on approximating convex, compact bodies in $d$-dimensional space $\Bbb R^d$ by convex polytopes. One of the main results in this area is that under some mild ...
6votes
1answer
393views
Condition Number dependent algorithms for matrix operations
Using the Conjugate gradient method we can solve a linear system $Ax=b$, where $A\in\mathbb R^{n\times n}$ in time $O(n^2 \sqrt{\kappa})$, where $\kappa=\frac{\sigma_\mathrm{max}(A)}{\sigma_\mathrm{...
- 958
5votes
1answer
166views
Finding $k \times k$ rectangle in a matrix with maximum sum
Given an $n \times n$ matrix $A$ with $0-1$ entries, I want to maximize $\sum\limits_{i \in I, j \in J} a_{ij}$ subject to $|I| = |J| = k.$ I expect the problem to be NP-hard, so I want a polynomial ...
- 299
2votes
0answers
80views
Confusion with the definition of Online Set Cover
I am confused on a technicality on how Online Set Cover is defined. One way to define it is: We are given a collection of sets $\mathcal{S}$ upfront, and in each time-step an element arrives to be ...
- 649
1vote
1answer
66views
$k-$median problem and filtering technique Lin and Vitter
I read a paper from Tardos et al. about $k-$medians in metric space problem: Given $N$ as set of points in metric space with distance function $c_{ij}$ for each $i,j\in N$, demand $d_i$ for each point ...
- 210
0votes
1answer
78views
An inequality about median of points in higher dimensions
Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\...
0votes
0answers
103views
Examples of Gaussian randomized algorithms
I've been thinking about algorithms of the form where a quantity $c$ can be viewed as the expectation of some estimator (random variable) $X$ and the expectation is taken over some multivariate ...
- 111
1vote
0answers
118views
Are there good analogues to Sparsest Cut/Balanced cut for vertex separators instead of edges cuts?
Most problems about cutting graphs into roughly equal parts such as Sparsest cut, Graph partition, Balanced Cut, etc are based on minimizing the size of an edge cut. Even if all of those problems are ...
1vote
0answers
60views
Find the minimum cost spider joining a root to some leaves
A spider is a tree with at most one vertex of degree greater than 2. This vertex is called the head of the spider. I am interested in the following problem: We are given an undirected graph $G = (V,E)$...
- 649
0votes
0answers
55views
Multi-dimensional 0-1 Knapsack problem with a high number of dimensions
I would like to solve a multi-dimensional 0-1 Knapsack problem, by looking for approximation algorithms with constant approximation ratio if possible. Here the particularity is that the number of ...
- 113
