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Questions tagged [approximation]

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

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An LP with two covering constraints - how to round

I came across an LP with two covering problems, and I wonder how to find a good approximation. For the relevant part of the LP: We have a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
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1answer
29 views

Non-existence of approximation algorithm for the knapsack problem

I am working on the following exercise: Prove that if $P \neq NP$, there cannot exist an approximation algorithm $A$ for the knapsack problem (KP) such that $\exists k \in \mathbb{N}, \forall I \in S: ...
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Randomized algorithm to compute cover radius?

I am self-study the book "Geometric Approximation Algorithms" by Sariel Har-Peled. And I stuck on a problem and don't know how to start it. Let $C$ and $P$ be two sets of point in the plane , such ...
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What are the current state of art approximation algorithm for NP-Hard problems? [closed]

I came cross some works try to use deep learning to approximate NP-Hard https://arxiv.org/pdf/1810.10659.pdf Though the paper seems to have very good results but based on the citations. I'm quit ...
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1answer
29 views

Shortest hamiltonian path for different dimension points

The shortest Hamiltonian path (solution) for a set of points in $\mathbb{R}^k$ (in Euclidean space) changes subject to $k$. For example if for $k=1$, the shortest Hamiltonian path will be the sorted ...
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1answer
43 views

What is an approximation factor for the Greedy Motif Search algorithm?

What is approximation factor for the Greedy Motif Search algorithm? I couldn't find an answer to my question except for the fact that the algorithm has a unknown aproximation factor. I'm not a native ...
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1answer
47 views

Is there an algorithm to overapproximate a context free grammar by a regular expression?

I understand that a context-free grammar is strictly powerful than a regular expression in that a context free grammar can represent any regular language, but not all context free languages can be ...
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1answer
40 views

MAXSAT approximation

We have been studying a 1/2-approximation for MAXSAT which runs in expected polynomial time, by randomly assigning True/False to each variable and repeating until we reach an assignment with at least ...
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Definition of unbounded approximation ratio

Suppose that there is a specific instance of a graph for which the approximation ratio of an algorithm polynomially increases with the number of nodes of the graph, say the approximation ratio is $n^2$...
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Weighted-Set Cover Approximation

So in the weighted-set cover problem, I need to determine the minimum weight cover. My algorithm calculates the efficiency for each set: ...
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1answer
27 views

Algorithm to pick elements in one array that sum to a setpoint, and corresponding elements in other array to average to a setpoint?

Note: The input file is an excel file. However, I am only looking for help with the algorithm as I can then code it in VBA. I need to scan both columns (shown below) to find any number of column 1 ...
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2answers
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Vertex cover of bipartite graph

A vertex cover is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A minimum vertex cover is a vertex cover with minimal cardinality. From codeforces, ...
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Delivering to two or more locations in one go while respecting deadlines?

Assume that I have a business where people can place product orders. Each order must be delivered within a time limit, say $x$ minutes. I need 15 minutes to make each product. However, multiple ...
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1answer
50 views

Clarification on NP-hardness and hardness of approximation results for set cover?

I'm not familiar with complexity theory at all so please correct me if I make any incorrect statements. I am wondering what is the hard case of set cover? My understanding of NP-hardness is that it ...
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1answer
53 views

Approximation algorithm for weighted set cover, using multiplicative weights

It is known that the problem of fractional set cover can be rephrased as a linear programming problem and be approximated using the multiplicative weights method, for instance this lecture note shows ...
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1answer
56 views

Approximation ratio of greedy algorithm for makespan

In the course notes for Stanford MS&E-319: https://web.stanford.edu/class/msande319/lec1.pdf Lemma 5 is given as: The approximation factor of the modified greedy [scheduling] algorithm is 4/3. ...
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1answer
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Job scheduling approximation

In the course notes for Stanford MS&E-319: https://web.stanford.edu/class/msande319/lec1.pdf Lemma 5 is given as: The approximation factor of the modified greedy [scheduling] algorithm is 4/3....
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38 views

invariant of bin packing

We are given an array of integers and a number K. We need to pack these integers into bins. The condition is that we have to use exactly K number of bins and each bin should have equal capacity. We ...
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2answers
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Find maximal subgraph containing only nodes of degree 2 and 3

I'm trying to implement a (Unweighted) Feedback Vertex Set approximation algorithm from the following paper: FVS-Approximation-Paper. One of the steps of the algorithm (described on page 4) is to ...
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Show that for each even n, there exists a graph with n vertices, such that the 2-approx VC alg returns a VC which is exactly twice the Minimum-VC

Question: Show that for each even n, there exists a graph with n vertices, such that the ALG(algorithm) returns a vertex cover which is exactly twice the size of minimum vertex cover. Define ALG: ...
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1answer
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Find the smallest set of strings which “covers” a given set of strings (coverage = containing as substring)

Let $S$ be a finite set of strings and $0 < k\leq l$ integers. We want to find the smallest set of strings $T(k,l)$ for which the following holds: $\forall t \in T(k,l): k \leq |t| \leq l$ $\...
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44 views

Knapsack approximation algorithm (weights scaling)

I am trying to prove the following Knapsack approximation algorithm, the problem definition: Input: A set $S$ of $n$ objects that contains weights and values: $w_1,w_2,\ldots,w_n$ (weights) $...
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1answer
157 views

Approximating the biggest acyclic subgraph of a given weighted digraph $G= (V,E)$

The base scenario is this: We're given a weighted, directed graph $G= (V,E)$, and are tasked to find an approximating algorithm which returns a digraph $G' = (V,E')$ (i.e. which only deletes edges) so ...
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1answer
94 views

Is this partitioning problem NP-complete?

I have a sequence of points $(x_1, \ldots, x_n)$ and a function $f$ that maps every consecutive subsequence (ie. of the form $(x_i, x_{i+1}, \ldots, x_j)$) to a real number. I want to split this ...
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1answer
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Minimal edge cover of the hypergraph

We know that minimal edge cover for the normal graph is polynomial time solvable. Is it also true for hypergraph?
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1answer
44 views

set cover to edge cover

I want to find set cover of this problem. I have sets, each of cardinality 3. I want to find set cover. This is what I am doing. Treat each set as an edge, which is incident on each of its element. I ...
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1answer
72 views

minimum cardinality maximal matching of graph

How to find minimum cardinality maximal matching? I tried that pick a edge from highest degree vertex remove other edges from same vertex and so on.
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1answer
74 views

Longest-path in a graph, where the path should be 'straight'

Is there any existing work done on finding paths that are geometrically straight? I encountered a problem where I'd need to find the longest straight(-ish) path in a web of connected nodes, each of ...
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1answer
20 views

Is there an FPRAS for the number of min st cuts in general graphs?

Provan and Ball [1] showed that the problem of counting the number of minimum st cuts is #P-Complete. What is known about the problem of approximating the number of min st cuts? Is it possible to get ...
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1answer
81 views

Max flow and Matching problem

Where can i find a list of problems reducible to max flow and matching problems. I need such examples to learn and practice .
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Relaxations for MILP with logical constraints

I have an LP with a (non-fixed) number of logical constraints in the form of $X_1 \rightarrow X_2$ (where $X_1$ and $X_2$ are linear functions inequalities of the $n$ input variables). To express ...
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Johnson-Lindenstrauss and k-means

I have a question about Johnson-Lindenstrauss and k-means. I m study a resource that explain a link between Johnson-Lindenstrauss and k-means. From what I understand, Johnson-Lindenstrauss helps us ...
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2answers
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set cover where only certain special subsets are allowed

I am trying to solve a problem which turns out to be a form of the set cover problem. I've implemented the greedy Set cover approximation algorithm for set cover, but it turns out to not be accurate ...
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1answer
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Vertex Cover approximation algorithm

I have an algorithm that solves the Vertex Cover problem. The algorithm is Repeat while there is an edge: Arbitrarily pick an uncovered edge $e = uv$ and add $u$ and $v$ to the solution. ...
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1answer
36 views

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
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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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1answer
36 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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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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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
45 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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1answer
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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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Maximum-minimum-satisfiability [closed]

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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1answer
34 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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1answer
203 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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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
31 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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1answer
27 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
36 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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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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$k$ -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 ...