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Questions tagged [approximation]

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

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what is a shearlet/shearlet transform and how can i use it?

from what wikipedia says shearlet is a successor to wavelets(whatever that is) and that they are extremely good at representing complicated data efficiently. But neither wikipedia nor other articles ...
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1answer
12 views

Approximation factor shorthand clarification

I'm starting to dabble in the world of approximation algorithms and had a question about the convention many papers will use when talking about the approximation factor. I know that an approximation ...
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20 views

Determining a value in an algorithm on its first run

I was given the following algorithm as a solution to one of my problems. However, I am baffled at understanding how c' is ever initialized. The first if statement can never be reached, as c' is never ...
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95 views

Proving there is no polynomial algorithm for independent set

I need some guidance in an assignment I'm doing. I'm at complete loss, he says the the MAXIMUM INDEPENDENT SET problem is NP-hard and then asks me to prove that there is no polynomial time for the ...
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23 views

Sort-of interval scheduling

The Problem I have a set of sets of time intervals (hour, minute, day of the week). I want to select exactly one interval from each of set, and I want to minimize... the number of pairwise ...
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1answer
17 views

Defining Gap Problems

I recently started studying about approximation problems in the complexity class that I'm taking. I feel like some of the definitions in this subject presented in my course and that I came across ...
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75 views

Squared euclidean distance high probability bound

I wish show squared Euclidean distance between $x$ and $y$ (they are 2 point uniformly at random in $[-1,1]^d$ ) is at least $(1-\epsilon)cd $ with probability $e^{\Theta \epsilon^2d}$ with $c<1$. ...
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32 views

Amplification for Randomized Algorithms

I'm trying to show Amplification works for randomized algorithms, and for randomized approximation algorithms. Amplification for randomized algorithms: Given a randomized algorithm with time ...
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+50

Maximum-minimum-satisfiability

In MAX-SAT, we are given a formula and want to maximize the number of satisfied clauses. I.e., given a formula $\phi = c_1 \cap \cdots \cap c_n$, where each $c_i$ is a disjunction, we want to find the ...
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29 views

Variant of an approximation algorithm for vertex cover

Here is an approximation algorithm that finds vertex cover of a graph. ...
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35 views

Family of graph with Approximation ratio = 2

My question today is about the approximation algorithms. Well, for Approx-Vertex-Cover problems , we know we can get ratio of 2 just by picking an edge and taking 2 endpoints of the same and ...
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1answer
60 views

Christofides algorithm (by hand)

I am following this algorithm example: https://en.wikipedia.org/wiki/Christofides_algorithm#example The graph: [![enter image description here][1]][1] ...
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17 views

Converting an approximate algorithm for the minimization to the maximization form

I have a $\rho$-approximate algorithm for a minimization algorithm, where the objective is to minimize $O$ ($\rho \geq 1$ is some constant), such that the algorithm's solution is always within $\rho$ ...
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1answer
22 views

About a pre-processing step for primal-dual weighted set cover problem

I was reading the paper titled "Primal-dual RNC approximation algorithms" by Rajagopalan and Vazirani. I have a problem of understanding the Lemma 4.1.1. They present a dual fitting based algorithm ...
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19 views

The notion of PAC in approximation algorithms

In computational machine learning, the notion of Probably Approximately Correct means that (generally speaking) we can find (or "learn") with a high probability a function which has "low error". Is ...
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1answer
27 views

Knapsack-like problem oriented on contiguous selection

Problem inputs is an ordered array $A = [a_1, a_2, \ldots, a_n]$ with $\text{weight}(a_i) = w_i$. We define a $subgroup$ of $A$, denoted by $B$, whose elements' indices are continuous in integer. For ...
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33 views

Different properties of Heavy-Hitters and Count-Min Sketch algorithms?

I'm currently using the Heavy-Hitters algorithm as described here and I'm wondering what if any space, time, accuracy, or real-world performance differences I would see if I were to switch to an ...
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41 views

$k$ -center with outliers - but the points are on a line

The classic $k$-center with outliers problem is NP-hard and there exist approximation algorithms to solve it. However, what if we assume that the input point are on a line, rather than in an ...
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1answer
33 views

Is deep learning appropriate to approximate dynamic programming problems?

I have a problem which can be completely solved using dynamic programming, but in a very intractable way (On^4, where n is around 1000). I won't get into the details of the problem since it's a bit ...
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1answer
54 views

A question about a variant of a knapsack problem

I have the following problem: Let $q_1,\cdots,q_k$ be natural numbers $> 0$, $q := \sum_{1\le i \le k}{q_i}$ and $s_1,\cdots,s_k$ be positive $>0$ real numbers, and $S$ be a positive real number....
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69 views

Hardness of approximating Minimum Cardinality Exact Cover

The Minimum Cardinality Exact Cover (MCEC) problem is just like set cover, but the output sets must be disjoint. Formally, given a collection of subsets $S$ of a finite set $U$, the problem asks for ...
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32 views

Are there matching upper bounds?

When reading about approximation algorithms, I often find the terms "matching lower bounds". As I understand, these means to provide examples where the approximation algorithm matches the proved ...
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1answer
182 views

Big-O / $\tilde{O}$ -notation with multiple variables when function is decreasing in one of its arguments

Say we have an algorithm that takes an input a triple ($X$, $A$, $\epsilon$), where $X$ is a sequence of $n$ bytes, of which the algorithm might query only a subset, and $A$ and $\epsilon$ are ...
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1answer
63 views

When to terminate search for path in an infinite grid

I'm learning shortest path algorithms like Dijkstra's, BFS, etc. I understand on a 2D finite grid there are boundary conditions (i.e. size of the grid) that help terminate the algorithm and keep it in ...
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What would be a generic strategy to proof $\alpha$-approximation algorithms?

I am studying combinatorial optimization (mainly problems that can be represented on graphs and 0-1 IP optimization) and have come across many $\alpha$-approximation algorithms for certain problems. ...
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1answer
21 views

Ensure groups of four 3-tuples have 9 unique numbers

Note: I know the numbers are arbitrary, but this problem about this size has practical implications. It is an applied algorithm problem. Suppose you have 200 bins. Each bin would be very happy to ...
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9 views

Why does economising the power series give me more error?

Suppose I want to economise $\sin x$ with the following taylor series: $$P_{2n-1}(x) = \sum^n_{i=1}(-1)^{i-1}\frac{x^{2i-1}}{(2i-1)!}$$ for the interval $[-1, 1]$ for $P_5(x)$. For $P_5(x)$, I have ...
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28 views

How Does Aproximate Reinforcement Learning Reduce State Space?

I was following reinforcement learning lecture from "CS188 Artificial Intelligence, Fall 2013". Here is the slide: In the video, the lecturer says with approximate reinforcement learning we store ...
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An approximation for the sum of k largest elements of n-sorted arrays?

Suppose we want to find the sum of the $k$ largest elements of $n$-sorted arrays. All arrays are containing $k$ elements. All elements are between 0 and 1, and the the sum of all elements in array $i$...
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1answer
90 views

Equivalent Colorings of Graphs

Call two proper graph colorings equivalent if one can be obtained from the other by a permutation of the colors. In other words, they are the "same" coloring. I'm interested in finding proper non-...
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1answer
41 views

On FPTAS and many one parsimonious reductions

We have two $NP$ complete problems $\Pi_1$ and $\Pi_2$. Suppose $\Pi_1\rightarrow\Pi_2$ be a many one parsimonious reduction. If $\Pi_1$ has an FPTAS then does $\Pi_2$ also have? If $\Pi_2$ has an ...
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20 views

Approximation factor for graph problems

I am attempting to figure out how to use approximation factors to determine the answers an algorithm can return for graph problems. For example, if a graph G actually has a maximum clique with 13 ...
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K-approximation algorithms [duplicate]

I am attempting to understand exactly how the terminology for k-approximation algorithms work, from when k is known and determining what k is. If we assume that a vertex cover in a graph G is of size ...
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1answer
56 views

Compute e^x given starting approximation

I have been looking for an algorithm to calculate $e^x$ to arbitrary precision (millions of digits). The best way I know is to use Taylor polynomials. However, the Taylor polynomial algorithm seems ...
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1answer
356 views

What does Arora mean by 'computational history'?

In Arora's paper, he wrote, Papadimitriou and Yannakakis also noted that the classical style of reduction (Cook-Levin-Karp [41, 99, 85]) relies on representing a computational history by a ...
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38 views

Spanning tree with equally separated edge weights

I have a fully-connected graph $G=(V,E)$ with edge weights $w(v)\in\mathbb{R};v\in V$ and I need to find a spanning tree $T=(V_t\subseteq V,E_t\subseteq E)$ where the set of edge weights in the tree ...
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subgraph that verify certain condition

Given an undirected graph $G=(V,E,p,c)$ and a positive integer $k$, $p: V \longrightarrow R^+$ which associates a positive weights $p(v_i)$ to every node $v_i \in V$, and $c: E \longrightarrow R^+$...
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How to find a Graph Embedding given a metric space? [closed]

I am interested to learn more about Topological graph theory and Graph Embedding. Assume I have a metric space, $d \colon M \times M \to \mathbb{R}$ and a graph, $G=(V,E)$. What is a rigorous way ...
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37 views

Competitiveness of bin packing with allocation constraint

I am working on a variant of the variable sized vector bin packing problem that is subject to an allocation constraint, where items can be put in a few particular bins only, not any bins. This item-...
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25 views

Hardness of approximation for online algorithms

Similar to the theory of hardness of approximation for (offline) approximation algorithms, has there been any work done on proving hardness guarantees for online algorithms? Theoretical lower bounds ...
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35 views

How does one find the maximum of $\{ \eta \in \mathbb R : |h(\eta)| \leq \epsilon \}$ using line search?

Assume I have a function that is computable in polynomial time $h(\eta)$ where $h: \mathbb R \to \mathbb R$. I am interested in computing the following supremum (approximately and in polynomial time ...
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difference between integrality gap and approximation ratio [duplicate]

What is the difference between integrality gap and approximation ratio? Any example to elaborate the difference?
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61 views

Is there an optimization problem on planar graphs which is APX-hard ?

I'm looking for a optimization problem on planar graphs which is APX-hard, which means that it doesn't admit a PTAS (approximation scheme). It would be even better is the difficulty of the problem ...
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1answer
71 views

An exact solution for biclique vertex-cover problem on a bipartite graph

The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most "k" bicliques (complete bipartite subgraphs). It has been shown that "Biclique Vertex-...
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1answer
56 views

Knapsack with non-overlapping rectangles

Given a large rectangle and a set $S$ of small rectangles each with a value $v_i$. My problem is to find a collection of rectangles $T\subseteq S$ to put in the large rectangle so as to maximize the ...
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41 views

Finding minimum number of bicliques that cover nodes on one side of a bipartite graph

Let $G=(U \cup V, E)$ denotes a bipartite graph. A biclique $C = (U, V)$ is a subgraph of $G$ induced by a pair of two disjoint subsets $U' \subseteq U$, $V' \subseteq V$, such that $\forall u \in U', ...
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1answer
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How does the strong form of the PCP theorem imply the inapproximability of Max-XORSAT?

Theorem 11.4 For any constant $\delta > 0$, every problem in NP has probabilistically checkable proofs of length poly($n$), where the verifier flips $O(\log n)$ coins and looks at three bits of the ...
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1answer
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Finding bicliques in a bipartite graph of minimum size

Let $G=(U \cup V, E)$ denotes a bipartite graph. A biclique $C = (U, V)$ is a subgraph of $G$ induced by a pair of two disjoint subsets $U' \subseteq U$, $V' \subseteq V$, such that $\forall u \in U', ...
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3answers
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How approximate sine using Taylor series

I need to approximate the sine function without internal libraries. I used Taylor series in 0 to solve this, but my program works for some values, but for others awful results. The program gets x ...
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Why gap preserving reduction is weaker than L-reduction?

In Vizirani's textbook says in page 332, Gap preserving reductions are weaker than their L-reductions [...] one of the motivations for the PCP theorem was that establishing an inapproximability ...