Questions tagged [approximation]

Questions about algorithms that solve problems up to some bounded error.

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108 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. ...
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1answer
47 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 $c_2$. ...
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39 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
28 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 ...
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2answers
110 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 ...
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2answers
40 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
138 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 ...
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1answer
22 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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17 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 ...
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1answer
62 views

Greedy algorithm for problem asking for solution of size *at most* $k$

Given an integer $k$ and a complete weighted bipartite graph with sides $A,B$ in which the weight of the edge $(i,j)$ is $c_{ij} \geq 0$, we want to find a set $S$ of at most $k$ edges that maximizes $...
2
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1answer
38 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 ...
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1answer
29 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 $1/4$?
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1answer
20 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
41 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 ...
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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. ...
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1answer
34 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 ...
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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 ...
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1answer
48 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 (...
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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
55 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 ...
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1answer
45 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
61 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
69 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 ...
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13 views

graph augmentation problem for Regional-based connectivity

Region-based connectivity: Given a graph $G(V,E)$, $\lbrace R_1,..., R_{k}\rbrace$ is the set of all possible regions (A subgraph with diameter $d$ or a subgraph centered at a node with radius $r$) ...
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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 ...
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1answer
75 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 ...
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1answer
43 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
61 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: ...
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1answer
39 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 ...
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1answer
30 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 ...
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1answer
116 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 ...
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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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20 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, ...
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5answers
306 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 ...
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1answer
40 views

Variant of greedy algorithm for vertex cover

Does the following approximation algorithm for vertex cover also have an approximation ratio of 2? Why? Why not? Input: $G = (V,E)$ Set $C \gets \emptyset$ and $E' \gets E$. while $E' \neq \emptyset$...
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1answer
58 views

Greedy algorithm for job scheduling

Consider the following greedy algorithm for Job Scheduling. For each new task, assign the task to processor with the shortest uptime. How to prove that this algorithm has an approximation ratio of ...
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0answers
36 views

Longest processing time rule on unrelated machines

I am trying to solve the job secheduling problem using the Longest processing time rule, that is: We seek to minimize the makespan by first sorting the jobs in decreasing order of processing time, ...
3
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1answer
87 views

Maximum coverage 1/2-approximation algorithm: why does the central lemma hold?

I am looking for an approximation algorithm for the Maximum Coverage problem and a proof of its approximation ratio. As approximation algorithm I use the greedy algorithm which chooses the set that ...
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1answer
314 views

Show that a $\alpha$-approximation algorithm is not a ($\alpha-x$) approximation algorithm for $x > $0

Suppose you have a system that consists of $m$ slow machines and $k$ fast machines. The fast machines can perform twice as much work per unit time as the slow machines. Now you are given a set of n ...
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19 views

Solving the multiple-choice knapsack problem for large input

I need to solve the multiple-choice knapsack problem for a very large input size ($\approx 10,000,000$). What is best way to practically do this? I've seen some papers describing FPTAS (=Fully ...
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0answers
34 views

Heuristic algorithm for the minimum weighted s-t cut with linear running time

To the best of my knowledge, the best algorithm for the minimum s-t cut in a weighted digraph is the Goldberg push-relabel algorithm with $O(n^{2}\sqrt{m})$ time complexity. I'm interested in solving ...
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1answer
25 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 ...
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2answers
76 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{...
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33 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 ...
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1answer
26 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 ...
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0answers
51 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
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0answers
108 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-...
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1answer
17 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 ...
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2answers
46 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 ...

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