Questions tagged [approximation]
Questions about algorithms that solve problems up to some bounded error.
473
questions
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maximal independent set on grid-based graph proof of approximation ratio
We have a G = (V, E, w), in form of a grid graph with a single diagonal line in each grid in form of below. Where w is the V weight.
We use a greedy algorithm that takes in each step maximum weighted ...
0
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1answer
18 views
maximal independent set on grid graph proof
I'm trying to figure out proof of maximum independent set from: this link. (1b part).
And I'm bit confused why exactly sum of $w(v)$ is less than or equal to sum of $w(v')$.
Shouldn't it be other way ...
1
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1answer
47 views
k-polynomial time approximation algorithm for set cover (k = max size of subsets)
Problem Definition: Given a universe set $U = \{1, 2, \dots, n\}$ and a collection of $m$ subsets $S_1, S_2, \dots S_m \subseteq U$, find the minimum collection of subsets that cover $U$.
I am ...
1
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1answer
28 views
Reduction from TSP to ATSP does not imply constant factor approximation algorithm
As I understand there is a constant factor approximation algorithm (e.g Christofides algorithm) for the symmetric TSP problem. This is however not the case for the asymmetric TSP problem (I am ...
1
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0answers
42 views
The correct way to calculate the performance of my approximate algorithm
I've got a question about how best to classify the performance of an approximate algorithm. I'm trying to find the 'correct' value of a graph problem instance whose cost function has an objective ...
1
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1answer
17 views
Can PTAS $\epsilon$ parameter be dependent on the algorithm input?
Let A be a PTAS algorithm with time complexity $O\left(\frac{1}{\epsilon}\right)$.
Let $n$ be the input of the algorithm A.
From Wikipedia:
The running time of a PTAS is required to be polynomial in $...
0
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1answer
25 views
How are matchings a lower bound for an approximate vertex cover?
I am reading Algorithms by Dasgupta et al and they mention maximal matchings as approximations for vertex cover.
They mention that the 2-approximation ratio is a lower bound. How is a maximal matching ...
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0answers
31 views
Polynomial variable of inapproximability after reduction
I proved the inapproximability of a problem that, given a multigraph $G = (V, E)$ and a set of vertices $U \subseteq V$ tries to maximize a score $f(U)$ whose value depends on the edges of the graph, ...
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2answers
19 views
Can you have an approximation that is higher than the optimal for a maximum value and a lower than the optimal for a minimum value?
I was reading this page on approximation ratios and the author says that for a problem looking for a:
maximum, an approximation algorithm will give us a value lower than this optimal maximum
minimum, ...
1
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1answer
21 views
Are disjoint edges the same as matchings?
I am reading Chapter 9 Approximation Algorithms of Dasgupts et al.'s Algorithm book for vertex cover approximation and they bring up the concept of matchings.
To support this, I am also watching ...
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1answer
95 views
Where does 1.3606 approximation ratio come from for vertex cover approximation?
I was watching a coursera video on Approximation algorithms and I understood the 2-approximation algorithm.
Later, the professor asks if we can do any better. The lecturer went on to say that ...
2
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2answers
79 views
Path graph partitioning, minimizing cut while minimizing maximum total node weight in each part
Suppose there is a path (linear) graph $G = (V, E)$ where $V = \{0, \ldots, n - 1\}$ and $E=\{(0, 1), (1, 2), \ldots, (n - 2, n - 1)\}$, with edge weights $w_e : E \to \mathbb{N}$ and vertex weights $...
2
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1answer
33 views
k-center problem: proof for Gon algorithm gives a 2-approximation
The $k$-center problem is where we a given a graph $G(V,E)$, an integer $k$, a distance metric $d$ and we want to find a subset $C\subseteq V$ (such that $|C|\leq k$) which minimizes the following ...
-1
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1answer
57 views
Clique is NP hard to approximate up to $n^{a}$ for some $a \in (0,1)$
Given that
$\mathsf{NP}=\mathsf{PCP}_{[\frac{1}{n},1]}\left(O\left(\log n\right),\left(O(\log n\right)\right)$,
show that it is NP-hard to approximate clique up to factor of $n^a$ for some $a \in (0,1)...
1
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1answer
25 views
Online approximation algorithm for median?
Is there a well-known or (relatively) easily-implementable streaming algorithm for approximating the median of the last, say, $k$ elements of a stream $c_1,c_2,c_3,\dots$?
The scenario is: I have a ...
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0answers
30 views
Cover a surface with disks of various size
I'm trying to find a cover of a surface using disks of variable sizes.
In the image, I want to cover the entire blue surface. The disks can go "out" of the blue surface and into the red part,...
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0answers
28 views
Is there an approximation algorithm for the three-person stable roommates problem?
While there's an algorithm for solving the stable roommates problem, I understand that the three-people-per-room version of that problem, sometimes called the "threesome roommates problem", ...
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0answers
135 views
Minimum Degree Spanning Tree Without Restricting Vertices Searched [closed]
I am currently self studying approximation algorithms from The Design of Approximation Algorithms (Williamson and Schmoys; page 50 here), specifically the minimum-degree spanning trees (MDST) problem. ...
2
votes
1answer
49 views
Tight approximation for the chromatic number of an arbitrary graph in polynomial space and time
I am looking for an algorithm for approximating the chromatic number of an undirected simple graph with $n$ vertices in $O(n^{c_1})$ time and $O(n^{c_2})$ space, for some constants $c_1$ and $c_2$. ...
0
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0answers
42 views
Most popular path in weighted cylic directed graph
Context
I have a graph $G=(V,E)$ with weighted edges, all weights are positive integers $w(e)\in\mathbb{N}\setminus\{0\}$. The weights represent the popularity/count of each edge, for example $w(e) = ...
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1answer
41 views
Approximate max weight path in directed graph
Context
This question is related to the fact one can't use Bellman-Ford to find max weight paths in directed graphs with cycles. The reason is that giving a new graph $\tilde{G}$ with negative weights ...
2
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2answers
123 views
Approximation of Set Cover
I wonder why do we say $\log n$ is the best possible approximation factor for Set Cover Algorithm? We already know there exists a 2-approximation algorithm for vertex cover, which is obviously better ...
2
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2answers
43 views
Prerequisites for studying parametrized complexity
Which areas of CS/Math should one have mastered before diving into parametrized complexity?
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6answers
186 views
What algorithm do computers use to compute the square root of a number?
What algorithm do computers use to compute the square root of a number ?
EDIT
It seems there is a similar question here:
Finding square root without division and initial guess
But I like the answers ...
1
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1answer
34 views
Find The “Best” Permutation of Inputs to Maximize Sum of Functions (or approximate “best”)
The Problem (in words)
I want to sort $N$ items where the value of item $i$ at position $p$ is given by the function $f_i(p)$. The "best" order for these items is the one that maximizes the ...
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0answers
18 views
Best algorithm / method to determine path length from noisy GPS points
i am analyzing a series of GPS points (with time stamps) which are noisy meaning have an accuracy of about 15 meter radius (sometime even more), and i need to extrapolate the distance the vehicle has ...
2
votes
1answer
42 views
2-Approximation algorithm for for messages across a cyclic network
Question
There are $n$ computers arranged in a cycle ($1,2,3..,n,1$), with undirected edges between adjacent computers. There are $m$ messages that need to be delivered. Message $i$ ($1 \le i \le m$) ...
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0answers
16 views
Additive approximation to bin packing
The bin packing problem is an NP-hard optimization problem that has many constant-factor approximation algorithms. I am looking for an additive approximation. I.e., given a set $I$ of items and bin ...
0
votes
1answer
48 views
Approximating longest path on graphs with average degree n/2
I have a graph with average degree $n/2$. How I can find an approximation algorithm for the longest path problem with factor $1/4$?
1
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1answer
29 views
Approximation algorithms for an instance of the Monotone circuit satisfiability
I have the following problem. Given a below boolean formula (of the type explained below) containing $n$ literals and two parameter $k$ and $l$, come up with a satisfying assignment of literals such ...
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0answers
44 views
Minimum Dominating Set
Consider a graph $G$ with minimum degree $d$, we know through sets cover, it's possible to find the one dominating set $S$ that covers $G$ such that $$S\leq O(\log n)\frac{n}{d}
$$ with high ...
1
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1answer
21 views
linear time nash equilibirum aproximations for two player zero sum games
I'm working on an AI for a game where I'd like the game where each player has hundreds of moves to select from and so the game matrix has 10s of thousands of entries. The game is however zero sum. ...
1
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1answer
43 views
Regula falsi with error in x-axis
I want the regula falsi to get x +- .0001 so that f(x) = 0. But all the implemetations I see get x so that f(x) +- .0001 = 0 which doesn't make much sense. (f(x) = x^3).
How do I stop the regula ...
2
votes
1answer
32 views
Why are $L$-reductions defined the way they are?
I was reading about $L$-reductions and there was one part in the definition that I thought was interesting. I wanted to know what motivated people who came up with it to have it included in the ...
2
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1answer
61 views
Approximation concerning Asymmetric TSP, Symmetric TSP, and Metric TSP
I always considered Symmetric TSP to be inapproximable in general, and thus by extension Asymmetric TSP as well. Once you add the condition of the triangle inequality however, you obtain Metric TSP (...
3
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0answers
47 views
Total weight of all spanning trees
Given a weighted simple undirected connected graph $G = (V, E, w:E \to \mathbb{R})$, let $\tau(G)$ be the set of all its spanning trees. Is there an efficient algorithm to determine or estimate with ...
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1answer
66 views
prove: max (w(E), w(E)) is a 1/2 approximation to the value OPT
Hey I would like to find a answer for b. for a look to the picture that is my answer for it. But I dont habe any Idea how i can solve this. Thank you guys. (I had to translate it to english maybe it ...
3
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1answer
53 views
Approximation factor preserving reduction
The definition of approximation factor preserving reduction from the book by Vijay V. Vazirani, page 365:
Let $\Pi_1$ and $\Pi_2$ be two minimization problems, an approximation factor preserving
...
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1answer
62 views
acyclic and disjoint union
I would like to find a prove of (a) so that the two E are acyclic and disjoint union and I dont unterstand b Could someone shed light on this problem, preferably spiced with some intuition?
Thanks, ...
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0answers
70 views
Greedy Probabilistic Algorithm for $Exact$ $Three$ $Cover$
I have a probabilistic greedy algorithm for Exact Three Cover. I doubt it'll work on all inputs in polytime. Because the algorithm does not run $2^n$ time. I will assume that it works for some but not ...
2
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0answers
48 views
Complexity of approximating a function value using queries
I am looking for information on problems of the following kind.
There is a function $f: [0,1] \to \mathbb{R}$ that is continuous and monotonically-increasing, with $f(0)<0$ and $f(1)>0$. You ...
5
votes
1answer
78 views
Maximize area of light with 4 light sources on a diagram of a room
Given a diagram of a room with obstacles in it (like walls or furniture), find the 4 best places to put omnidirectional light sources in it so the area that is lighted is maximized.
Here is a simple ...
1
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1answer
44 views
TSP 200-approximation, given $c(x,z)\le c(x,y) + 100\cdot c(y,z)$ for all nodes $x,y,z$
Input: complete, undirected graph $G=(V,E)$ and cost function $c$
Assume for all nodes $x,y,z \in V$: $c(x,z)\le c(x,y) + 100\cdot c(y,z)$
Find a 200-approximation polynomial time algorithm for the ...
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2answers
62 views
Coloring graph with constraints on assortativity
I have come across the following problem and I would love for your thoughts on an optimal solution or approximate calculations worth trying.
The formulation of the problem in the form of graph theory:
...
1
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1answer
45 views
Max independent set in planar graphs PTAS proof
I've been searching a few hours for a proof to Max independent set in planar graphs beeing in PTAS but I couldn't find anything, I'm searching for one without any reductions and I wonder if anyone ...
2
votes
1answer
34 views
Uncapacitated facility location problem using local search
I'm studying about UFLP using the book The Design of Approximation Algorithms Ch 9 starting page 233 (there is an electronic free edition), I ran into some unclear steps in the book and need some help ...
1
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1answer
118 views
Approximation algorithm question, clustering on n points
So the algorithm I thought of, is to iterate through the n points, centering a ball at each point, and keeping track of the point where we centered that encapsulated the most points. Then remove the ...
0
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0answers
21 views
Applying Polynomial Time Approximation Scheme (PTAS) on an Algorithm
I am trying to apply PTAS on an algorithm. I think that we apply PTAS on the running time equation of the algorithm. We use the term (1-ϵ) and (1+ ϵ) in the running time of the algorithm but I don’t ...
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0answers
21 views
Set Cover: Understanding the algorithm with an example
I am trying to follow the following link:
Solution for an example
They have provided solution for an example using the greedy algorithm. I have got following questions:
(1)Why start with Z, cost is 7, ...
5
votes
5answers
321 views
fast and stable x * tanh(log1pexp(x)) computation
$$f(x) = x \tanh(\log(1 + e^x))$$
The function (mish activation) can be easily implemented using a stable log1pexp without any significant loss of precision. Unfortunately, this is computationally ...
