Questions tagged [approximation]
Questions about algorithms that solve problems up to some bounded error.
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vertex cover of bipartite graph
A vertex cover of a graph 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.
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What is the source to understand Feedback edge arc set? [on hold]
What is the source to understand Feedback edge arc set?
I tried wikipedia and research papers any easy tutorial?
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0answers
40 views
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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0answers
46 views
Approximation Algorithms
Atleast give the reason for downvoting.I am crying.
vijay vazirani -approximation algorithms
I am trying to understand max sat approximation algorithm from https://doc.lagout.org/science/0_Computer%...
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1answer
45 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
31 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
47 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
42 views
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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1answer
37 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
72 views
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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0answers
41 views
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
52 views
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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0answers
27 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)
$...
3
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1answer
106 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
90 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
34 views
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
42 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
52 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
70 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
19 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 ...
2
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1answer
75 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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0answers
25 views
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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0answers
16 views
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
75 views
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
78 views
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. ...
2
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1answer
32 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
15 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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0answers
21 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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0answers
140 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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0answers
30 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
30 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
38 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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0answers
89 views
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
131 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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0answers
20 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
29 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
24 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
35 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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0answers
38 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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0answers
42 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
99 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
78 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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1answer
98 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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0answers
48 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
261 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
109 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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22 views
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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0answers
10 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 ...
