# Questions tagged [approximation]

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

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### 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 vote
134 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 ...
47 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 ...
460 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 ...
1 vote
65 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 $E' \neq \emptyset$...
384 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 ...
1 vote
44 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, ...
356 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 ...
1 vote
737 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 ...
1 vote
38 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 ...
53 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 ...
164 views

35 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 ...
72 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-...
406 views

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### 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 ...
121 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 ...
109 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
177 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
67 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 ...
55 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 ...
32 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 ...
1 vote
39 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) ...
158 views