Questions tagged [approximation]
Questions about algorithms that solve problems up to some bounded error.
426
questions
0
votes
1answer
17 views
Piecewise linear approximation for set of points
If we have a set of points $A$ and a known algorithm $Bestfit(A)$ to find
the best-fit straight line-segment through $A$, what would be a good algorithm to construct piece-wise linear segments (with ...
0
votes
2answers
42 views
Numerical Approximation in Java
I am trying to solve an equation which I believe cannot be done analytically, but can use a numerical approximation to get a result. The equation is:
$$\frac{2*\sqrt{\pi}*h*s*e^{m^{2}/(2*s^2)}}{\sqrt{...
-1
votes
0answers
19 views
How to prove that degree-4-IS si NP-Complete
I would like to have an idea to solve this kind of problem.
Let us call degree-k-IS the restriction of the problem of the independent set to graphs of degree
maximum $k$.
Show that the degree-4-IS ...
0
votes
0answers
32 views
Approximation Set Covering Problem
I am studying approximation Set Covering problem, from "Introduction to Algorithms by Thomas H. Cormen" book. What I cannot understand is why they use harmonic numbers to the proof? What harmonic ...
1
vote
1answer
23 views
3-Approximation for General position subset
I am currently studying for an exam and stumbled upon the following task:
Given the following problem:
Input A set of points $P \subseteq \mathcal{Q}^2$ and $k \in \mathbb{N}$
Question Find the ...
1
vote
0answers
43 views
Minimum-cut with balanced and limited number of nodes in each partition: Does this have an efficient solution or even a name?
I'd like to remove the minimum number of edges from an undirected unweighted graph to partition the nodes into an arbitrary number of connected components $S_1$, $S_2$,$S_3$,... $S_k$ while maximizing ...
2
votes
0answers
86 views
PTAS for Multiple Knapsack with Uniform Capacities, fixed number of Knapsacks
Consider the following problem:
We are given a collection of $n$ items $I = \{1,...n\}$, each item has a size $0 < s_i \le 1 $ and a profit $ p_i > 0 $. There are $m$ (a fixed number) of unit-...
0
votes
0answers
28 views
Scheduling jobs on a single machine - minimising the weighted sum of completion times
Consider the following problem:
there are $n$ jobs $\{1,...,n\}$, each has a processing time, $p_i$, a weight $w_i$, and an arriving time $r_i$. The goal is to minimise the weighted sum of completion ...
0
votes
0answers
32 views
Graham 1966 Minimum makespan scheduling
I am trying to understand how Graham's 1966 makespan algorithm results in 2-approximation. I have searched in google and I have found files, and videos on youtube about makespan scheduling, but again ...
0
votes
1answer
16 views
Compendium of approximation ratios, with narrower scope and more up to date than the NP-compendium?
I usually check the NP-compendium from Pierluigi Crescenzi, and Viggo Kann when I want to know the APX status and approximability results of a problem.
However, I understand that maintaining it is a ...
0
votes
2answers
42 views
Calculating match % and ranking according to that
I'm creating a website like where users will answer some yes/no questions set by me, up to them how many of those questions they want to answer. After a user submits his answer(s), he will be shown ...
0
votes
0answers
47 views
A variant of the knapsack problem
Consider the following variant for the knapsack problem: the input are disjoint sets of items $ T_1, T_2, ..., T_m$ (each contains items of a different type). Every item $i$ has a value of $v_i$ and a ...
8
votes
0answers
114 views
Compute the expected size of an approximation of vertex cover
Consider the following randomized approximation algorithm of vertex cover:
Input: A graph G = (V, E).
Output: A set $C_G \subseteq V$ a vertex cover of $G$.
The algorithm:
Set $C_G := \emptyset$.
...
2
votes
0answers
22 views
FPTAS algorithm to find flow at each link for multi commodity flow problem?
Given a graph $G$ and $K$ commodities to route from source to destination. I want to find, what is the maximum beneficial flow for each of the commodities and the relevant paths. I understand the ...
1
vote
1answer
556 views
Derandomization of vertex cover algorithm
I have the following randomized-algorithm for the vertex cover problem. Let $B_0$ be the output set:
Fix some order $e_1, e_2,...,e_m$ over all edges in the edge set E of G, and set $B_0=ā
$.
Add to ...
2
votes
1answer
69 views
Integrality gap and LP-rounding
I have a doubt about integrality gap.
If I know that there is no integrality gap for a given problem, i.e.:
$$\frac{\mathrm{OPT}(\mathrm{ILP})}{\mathrm{OPT}(\mathrm{LP})} = 1 \text{ (right?)},$$
...
3
votes
0answers
33 views
Estimating number of points in 1D space
There are some arbitrary-chosen points in 1D space. What needs to be found is the approximate number of them without counting all of them. It is possible to choose some coordinates (numbers) and for ...
0
votes
0answers
56 views
Doubt on integrality gap and LP relaxation
I have an exercise that tells me that, given a problem P (of which now I omit the description) there is no integrality gap between LP and ILP formulation of this problem, and for every fractional LP-...
9
votes
0answers
176 views
2D interval scheduling problem
Suppose I give you $n$ axis-aligned rectangles with a specified width, height, and x-position (of the left edge) $\{(w_i, h_i, x_i) \mid i \in \{0, \ldots, n - 1\}\}$, as well as a bound $(y_\mathrm{...
9
votes
2answers
223 views
Prove that the 2-approximation of a modified local search algorithm for max-cut is tight
Consider the following local search approximation algorithm for the unweighted max cut problem:
start with an arbitrary partition of the vertices of the given graph $G = (V,E) $, and as long as you ...
1
vote
0answers
18 views
ln(n) + 1 Approximation for Set Cover constructions
Set Cover Problem: Given a set $X$ and a collection of subsets $S_1, S_2, \ldots S_m \subseteq S$, we want to find the smallest cardinality of a set of $k$ elements $\{i_1, \ldots i_k \}$ such that $\...
5
votes
1answer
96 views
Minimum edge weight k-exact cover with triangle inequality
Suppose you have a weighted graph $G = (V, E \subseteq V^2, w \in E \to \mathbb{R}^+)$, where $w$ satisfies the triangle inequality $w(x, y) + w(y, z) \ge w(x, z)$. Suppose $k \in \mathbb{N}$ (in ...
4
votes
0answers
73 views
Approximation algorithms for indefinite quadratic form maximization with linear constraints
Consider the following program:
\begin{align}
\max_x ~& x^TQx
\\ \mbox{s.t.} ~& Ax \geq b
\end{align}
where $Q$ is a symmetric (possibly indefinite) matrix and the inequality is element-wise ...
0
votes
0answers
31 views
A heuristic for finding an edge cycle cover
I am looking to find a minimum list of cycles in a graph such that their union gives the list of all simple cycles in this graph.
In the example below, here are 4 simple undirected cycles: 1-2-3, 2-3-...
1
vote
0answers
64 views
Tight analysis for the ration of $1-\frac{1}{e}$ in the unweighted maximum coverage problem
The unweighted maximum coverage problem is defined as follows:
Instance: A set $E = \{e_1,...,e_n\}$ and $m$ subsets of $E$, $S = \{S_1,...,S_m\}$.
Objective: find a subset $S' \subseteq S$ such ...
1
vote
1answer
29 views
Reductions from non decision problems
I want to show a minimization problem $Y$ has no approximation factor of 1.36. To be more specific the problem $Y$ is the exemplar distance problem between two genomes. Could I reduce from the min ...
3
votes
0answers
40 views
Approximate algorithms for class P problems
As a part of my Algorithm course we studied Approximate Algorithms for NP-complete or NP-hard problems, e.g. "set cover", "vertex cover", "load balancing", etc. My professor asked us as an extra ...
3
votes
0answers
26 views
Rearrange items in order reduce fragmentation and reduce wasted space
I have a segment with some offsets at irregular intervals
There are items of various length inside. Items cannot be placed randomly. Instead, their left side must match some offset.
Items are free ...
0
votes
0answers
11 views
What are some counter examples for Load balancing problem?
I learnt about load balancing problem under approximate algorithms. I am learning about it, and studying methods to counter part it's non -solvability under P time. I am quite out of my examples, and ...
1
vote
0answers
26 views
Randomized version of the class $APX$?
Is there a class which is to APX what BPP is to P?
I'm looking for a definition that is like the following:
"For $r > 0$, an $r$-RPCA (randomized polynomial-time constant-factor approximation) ...
3
votes
1answer
80 views
Optimal solution for Weighted points problem
Problem:
Fix a constant $k$. Given a set of $2d$-dimensional points $N = \{N_1, N_2, N_3, \dots, N_n\}$, each associated with an arbitrary weight, find a set of points $X = \{X_1, X_2, X_3, \dots, ...
1
vote
0answers
37 views
About Steiner tree problem in graphs
In the
paper (p. 3)
and the
slides
presents the formulation of the Steiner problem on graphs
via so called Steiner cuts.
But according to the definition, the number of Steiner cuts and so the ...
1
vote
2answers
88 views
Can non-metric TSP be approximated within some non-constant value?
It is known that metric TSP can be approximated within some constant value, such as 3/2 through Christofides' algorithm. It is also known that non-metric TSP cannot be approximated within some ...
0
votes
0answers
39 views
Greedy algorithm for feedback vertex set / greedy algorithms vs local ratio in general
A greedy algorithm for finding a minimum feedback vertex set
is to pick and remove a vertex with minimum $w(v)/\delta_H(v)$, where $H$ is the current graph, until there are no more cycles left. (...
0
votes
0answers
54 views
What is the approximation for odd cycle transversal?
What is the best approximation for odd cycle transversal? (on general graphs)
Sorry if this is easily found everything I found about odd cycles is about paramaterized complexity and kernels
0
votes
0answers
16 views
Algorithmic question: distribute balls, optimise for balancing (i) weights (ii) probabilities of picking balls
I have an algorithmic problem that requires some lengthy explanation, which follows below.
tl;dr: distribute balls with weights among bags, optimise for balancing both (i) the weights between the ...
2
votes
1answer
31 views
How to use c-gap problems to prove inapproximability?
Suppose there is a specific set function with some properties - $f=2^V\to \mathcal{R}$. It is known that the following problem is NP-Hard: Find $S\subseteq V, |S|\leq k$ such that $f(S)$ is maximized....
3
votes
1answer
27 views
Hardness of approximation statement clarification?
In the paper I'm reading, there is a hardness of approximation result for an algorithm proved using a reduction to set cover. Roughly, the claim states that if there existed an algorithm that solved ...
0
votes
1answer
117 views
Difference Between PTAS and FPTAS [duplicate]
According to this link:
Polynomial Time Approximation Scheme (PTAS) is a type of approximate
algorithms that provide user to control over accuracy which is a
desirable feature. These algorithms ...
3
votes
1answer
53 views
Is this variation of set-cover NP-hard to approximate?
The classic set-cover problem is described as follows:
Let $S = \{s_1, ..., s_n\}$ be a target set, and let $\Lambda = \{A_1,
..., A_m: A_i \subset S\}$ be a collection of subsets of $S$. The
...
4
votes
0answers
61 views
Peculiar MCMC sampling problem
I have two random variables, X and Y, and Y is a positive real number. I can sample from $p(y|x)$, but I need to sample from $p(x)$, which I know to be proportional to $\frac 1 {E[y|x]}$. I could ...
3
votes
0answers
57 views
Alternative criterion for approximate maximum-weight perfect matching algorithms [closed]
Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not the factor between the approximate and exact weight sum achieved by each solution, but ...
1
vote
1answer
61 views
Analysis of an approximation claim
Consider the load balancing problem on two machines. Thus we want to distribute a set of $n$ jobs with processing times $t_1,...,t_n$ over two machines such that the makespan (maximum of the ...
1
vote
0answers
43 views
An LP with two covering constraints - how to round
I came across an LP with two covering problems, and I wonder how to
find a good approximation. For the relevant part of the LP: We have
a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
3
votes
1answer
32 views
Non-existence of approximation algorithm for the knapsack problem
I am working on the following exercise: Prove that if $P \neq NP$, there cannot exist an approximation algorithm $A$ for the knapsack problem (KP) such that $\exists k \in \mathbb{N}, \forall I \in S: ...
4
votes
0answers
56 views
Randomized algorithm to compute cover radius?
I am self-study the book "Geometric Approximation Algorithms" by Sariel Har-Peled. And I stuck on a problem and don't know how to start it.
Let $C$ and $P$ be two sets of point in the plane , such ...
0
votes
1answer
49 views
What are the current state of art approximation algorithm for NP-Hard problems? [closed]
I came cross some works try to use deep learning to approximate NP-Hard
https://arxiv.org/pdf/1810.10659.pdf
Though the paper seems to have very good results but based on the citations. I'm quit ...
2
votes
1answer
41 views
Shortest hamiltonian path for different dimension points
The shortest Hamiltonian path (solution) for a set of points in $\mathbb{R}^k$ (in Euclidean space) changes subject to $k$.
For example if for $k=1$, the shortest Hamiltonian path will be the sorted ...
2
votes
1answer
55 views
What is an approximation factor for the Greedy Motif Search algorithm?
What is approximation factor for the Greedy Motif Search algorithm?
I couldn't find an answer to my question except for the fact that the algorithm has a unknown aproximation factor. I'm not a native ...
3
votes
1answer
64 views
Is there an algorithm to overapproximate a context free grammar by a regular expression?
I understand that a context-free grammar is strictly powerful than a regular expression in that a context free grammar can represent any regular language, but not all context free languages can be ...
