Questions tagged [approximation]
Questions about algorithms that solve problems up to some bounded error.
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Inapproximability of graph problems on a restricted setting
I am considering the following problem $\mathcal{P}$.
$\mathcal{P}$: Given an undirected graph $G$, and an integer $k$, find a set of vertices $S \subseteq V(G)$, with $|S| = k$, such that the number ...
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Is there an FPTAS for 3-way number partitioning?
The maximization problem of the 3-way number partitioning reads as follows:
given $n$ positive integers, partition them into 3 subsets such that the smallest sum is as large as possible. It is known ...
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Hardness of multiplicative vs. additive approximation
Chlebik and Chlebikova prove that the problem "maximum 3-dimensional matching" is NP-hard to approximate within a multiplicative factor of $95/94$.
This means that, unless $P=NP$, there is ...
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What is a term for a problem that is hard to approximate within a factor $c$?
Let $f$ be a maximization problem. If there is a reduction from SAT to the following problem: "given an integer $c$, decide if there is an $x$ for which $f(x)\geq c$", then $f$ is NP-hard. ...
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An approximation variant of the halting problem
It always has been bugging me that we (humans) know pretty easily when most programs we write halt or not, but the halting problem is still undecidable.
I have just thought of a variant approximation-...
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Universal approximation bounds of the form $\|f(x)-\hat{f}(x;w)\|\leq \varepsilon \|f(x)\|$
It is known that for every $\varepsilon>0$ there is an appropriate neural network architecture, such that one can approximate any continuous function $f:[0,1]^n\to[0,1]^m$ by the neural network ...
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$\epsilon$-approximation Sub-linear time monotonicity testing
I have the following exercise I have been staring at for several hours to no avail.
Question:
Testing the monotonicity of a function - the case of bits: Given a function $f: [n] \rightarrow \{0,1\}$ ...
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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 ...
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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 ...
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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 ...
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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 $...
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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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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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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, ...
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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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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 ...
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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 $...
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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 ...
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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)...
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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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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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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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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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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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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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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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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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Prerequisites for studying parametrized complexity
Which areas of CS/Math should one have mastered before diving into parametrized complexity?
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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