Questions tagged [set-cover]

Set cover is a well-known NP-complete problem: given a collection of sets, find as small as possible a subcollection whose union is the same as the union of the entire collection.

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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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1answer
151 views

Given N sets of disjoint subsets, find a set of disjoint subsets such that it satisfies a criteria

Given a collection of sets $S_i$ of disjoint subsets $sub_i$ of a set $X$, find a set $A$ of disjoint subsets $asub$ such that each one of these subsets is subset or equal to at most one subset in ...
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2answers
978 views

How to enumerate minimal covers of a set

I have a set $S$ and a set $P = \{P_{1},...,P_{n}\}$ with $\bigcup P_{i} = S$. I want to find all the inclusion-minimal subsets of $P$ that are covers of $S$. What is the best algorithm for ...
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49 views

Upper bound on Set Cover Size

I am working on a related Steiner tree problem that I have reduced to Minimum Set Cover, but stumbled across this related problem and got stuck. Given an universe of $n$ elements $U = \{1,2,\ldots,n\}$...
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46 views

Boolean matrix / satisfiability problem [duplicate]

Let $M$ be an $m\times n$ matrix with all elements in $\{1,0\}$, $m >> n$. Let $\mathbf{v}_0, \ldots, \mathbf{v}_n$ be the columns of $M$. I want to find all sets of columns $S = \{\mathbf{v}_{...
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Given multiple families of sets, select a set from each such that the intersection of selected sets is a correspondence

Let $X = \{x_1, ... , x_k\}$ and $Y = \{y_1, ... , y_h\}$. A subset $S \subset X \times Y$ is a correspondence if : $\forall x, \exists y, (x,y) \in S$ and $\forall y, \exists x, (x,y) \in S$ That ...
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52 views

On the complexity of Unique Coverage Problem restricted to geometrical settings

I come to you with a problem I have been struggling with for the past few weeks. I would like to classify (complexity) a special case of Unique Coverage. To set the mood on, I will start by ...
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51 views

Smallest set non-disjoint with other given sets

Given a number of sets, what is the best algorithm to calculate the smallest set S such that S is not disjoint with any of the given sets?
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2answers
3k views

Vertex cover algorithms for directed graphs?

I've recently been working on a problem that I believe can be expressed as a vertex cover problem over a directed graph. Formally, I have a graph $G = (V,E)$ where $V$ is a vertex set and $E$ is a ...
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1answer
222 views

Proving NP hardness of maximum sum of means of a partition into k sets

I am trying to show the following problem is NP-hard and would like some help. Inputs: Integer $k$, and unordered set of $N$ numbers, $O$ Output: the $\max \sum\limits_{S_i \in S} \operatorname{mean}(...
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1answer
350 views

Counterexample to greedy solution for set cover problem

I am looking for the answer to the exercise 1-6 in "The Algorithm Design Manual" book. it is stated as follows: 1-6. The set cover problem is as follows: given a set of subsets $S_1,\dots,...
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1answer
582 views

Is the exact cover problem NP-hard when there is a restriction on the size?

The exact cover problem with restrictions on the size is: Input: Given a set $U=\{1,2,\ldots,n\}$ and a collection of $C$ of subsets of $U$. Question: Is there a subcollection $C^\star$ of $C$ such ...
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1answer
84 views

Selection over combinatorics that satisfies a distribution

I'm having an exciting problem that I could not manage to find an optimized solution. I actually have no idea if the problem is already known or not. Here is the problem : Consider a list of M ...
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1answer
762 views

Set cover problem with sets of size 2

I have a question about the Set Cover problem: If I get a universe $U$, and $m$ subsets of size exactly $2$, and an integer $k$. Is this problem is still NP-C or I can solve it on a polynomial time? ...
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1answer
23 views

Trouble to understand the proof of greedy algorithm for set cover

Problem definition: Given a universe $U$ of $n$ elements, a collection of subsets of $U$, $S = \{S_1,..., S_k\}$, and a cost function $c: S \to Q^{+}$. Find a minimum cost subcollection of $S$ that ...
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1answer
24 views

Efficient cardinality of set overlap relation

Assume that we have a set S of sets s. Every pair (s,s') in ...
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1answer
67 views

Triangles covering all vertices of a tri-partite graph

This question is an extension of this one: Min path cover for a three-layer graph with all paths traversing all layers. I'm designing fictional fruits. Each fruit has three attributes; color, taste ...
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1answer
453 views

Maximum weighted disjoint set union

I would like to know whether the following problem is a standard problem that has been considered in the research literature. I performed some searches, which have not produced results. I call this ...
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1answer
66 views

Given a set C of set P of sets R, find the smallest set S such that at least one subset R from each P is a subset of S

Given a universal set of elements $\mathbf{U} = \{a_1, a_2, .., a_n\}$, a set $\mathbf{R} = \{a_i\} \subset \mathbf{U}$ where $i ϵ \{1, .., n\}$, a set $\mathbf{P} = \{R_1, .., R_m\}$ and a set $\...
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1answer
807 views

What is Unique Coverage Problem?

I am trying to understand the problem statement and the approximation algorithm for the Unique Coverage Problem from these notes. I have refined the question based on the comments provided. Thanks to ...
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1answer
544 views

Problem with understanding the lower bound of OPT in Greedy Set Cover approximation algorithm

From what I know of analyzing and designing approximation algorithms, we need to find a lower bound on the optimum (in the case of minimization). For example if our solution is greedy ($SOL_G$) and if ...
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1answer
47 views

Find the n best choices while maximizing value

Given: A list of slots (A, B, C, ...) Every slots supports a list of choices (C0, C1, C2, C3...) Every choice has a value All slots must be filled with at most n different choices. The sum of the ...
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1answer
986 views

Is time complexity of the greedy set cover algorithm cubic?

I claim that the greedy algorithm for solving the set cover problem given below has time complexity proportional to $M^2N$, where $M$ denotes the number of sets, and $N$ the overall number of elements....
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1answer
390 views

Is Finding A Hitting Set of Size n/2 NP-Hard?

In Hitting Set problem we are given a collection E of subsets of V and we want to find smallest subset H of V which intersects (hits) every set in E. In decision version of the problem, we are asked ...
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1answer
47 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
79 views

NP completeness proof of sensor selection problem

There are $n$ points in a plane. The decision problem is to identify whether there exists a set $S$ of $k$ or less points from the $n$ points such that all $n$ points are at most $d$ distance from ...
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1answer
735 views

Reverse cartesian product matching all given rows

I´m looking for an efficient algorithm that will find reverse cartesian products. Mathematically, given $S \subseteq T^n$, I want to express $S$ as a union of sets $A_{i,1} \times A_{i,2} \times \...
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1answer
272 views

Verifying if a greedy solution is optimal for a specific instance (Set cover)

Say I have an instance of the Set Cover problem, and use the typical greedy algorithm to obtain a solution. Is there an efficient way of verifying if, for that particular instance, the given solution ...
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1answer
134 views

Optimal covering of 2D matrix elements given spatial constraints

I have a particular problem I need to solve, but I'm not sure how to classify the problem or pick the right algorithm to solve it. I'm hoping someone here can lead me in the right direction. I've ...
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1answer
85 views

Finding a subset of triplets of digits 0-9 such that each digit occurs 40 times in each position in the triplets

I am trying to generate a list of digit triplets to specify stimuli in an auditory (speech-in-noise) perception experiment. Each triplet has to have three different digits (i.e., no repetition within ...
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1answer
106 views

Approximation rate for Greedy Set Cover algorithm (RESOLVED)

Set Cover: Consider a set of points X and Si a subset of X. The goal is to get the minimum number of subsets Si such as all points in X are covered. An example is shown by figure bellow. In this case, ...
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1answer
76 views

Minimum covering problem formulation

Shouldn't I post this question on mathematics.stackexchange.com? Let be an airlaine company which has to affect its aircrew to several flights. We group som flights in subset, every flights of a ...
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1answer
217 views

Another version of the online set cover problem?

Here is a note about online set cover problem: we are initially given the $m$ sets, but we do not know which elements they contain. At any time $t$, we get a new element $e_t$ and learn which sets ...
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1answer
49 views

How can I convert a list with duplicates into a set for a reduction to the set cover problem?

I'm trying to come up with a reduction for a problem whose description is more or less identical to the first problem given here. Here's a condensed version of the problem: You're given a collection ...
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67 views

A Special Case of Set Cover Problem: Covering Nodes of Tree using Paths [closed]

Let $U$ be the set of elements and $S$ be the subset collections. There exists a tree $T$ that each node is corresponding to an element in $U$. And for every subset $s$ in $S$, $V(T) \bigcap s $ is a ...
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Variant of "Exact Cover by 3-Set "

Exact cover by 3-sets is 𝖭𝖯-complete: Instance: Given a finite set $X = \{x_1, x_2, …, x_{3n}\}$ of $3n$ elements and a collection $C = \{(x_{i_1}, x_{i_2}, x_{i_3})\}$ of 3-elements subsets of $X$; ...
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Find optimal starting point of recursive search, with solution known

An Exact Cover problem is commonly denoted as a matrix of 0s and 1s. Columns denote requirements, rows denote possible choices. The problem is solved when a set of rows was found such that, using just ...
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Complexity of solving fractional constrained set multicover

Recently, I've encountered the following problem: Given a collection of sets $S_1 \dots S_n$ of elements $e_1 \dots e_k$ with element $e_k$ denoted privileged, and a $k-1$-vector $r$, choose at most $...
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109 views

Is this a hitting set or set cover problem? [closed]

Define a universe $U$ containing $N$ elements. We are given $N$ sets, each of which is a set. For example, $U = \{1, 2, 3, 4\}$ and sets \begin{align} S_1 &= \{\{1\}, \{2, 4\}\}, \\ S_2 &= \{\{...
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117 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 ...
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0answers
45 views

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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159 views

Solving the Size-Constrained Weighted Set Cover Problem

I'm wondering if anyone has experience trying to solve a weighted set cover problem over the power set (i.e. all possible subsets) of an $n$-element ground set where the number of sets included in the ...
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2answers
716 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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164 views

Applications of Set Covering

I am interested in the applications for set covering. On Wikipedia and this thread, I read that it is used in antivirus programs, random testing in software, and personnel scheduling. Does anyone know ...
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103 views

How to enumerate all partitioning of a set to k-subsets of size at most b

I'm looking for an algorithm to generate/enumerate all possibilities for partitioning a set of size $n$ to $k$ non-empty subsets, each with size at most $b$. More specifically, given a set $V$ where $...
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0answers
44 views

Algorithms for MinSat when all literals are positive?

I am given a set $U = \{1, \cdots, n\}$ and a set of subsets $C_1, \cdots, C_k$ of $U$; here the sets are all the same size. The minimum set-cover problem of course is to find the minimum number of ...
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35 views

NP-hardness of maximum set cover with element-level submodular function

Consider the following generalization of maximum set cover problem: Given a collection $C$ of subsets of a finite set $S$. Find $C^{'} \subseteq C$ of cardinality $k$ that maximizes \[ \sum\...
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491 views

Is this a proof that SET COVER is not an NP-hard problem?

In this paper, Karpinski and Zelikovsky introduce the SET COVER and the $\epsilon$-DENSE SET COVER problems as follows: Set Cover Problem. Let $X = \{x_1, \ldots, x_k\}$ be a finite set and $P = \{...
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1answer
27 views

Will this reduction of Exact Cover into Subset-Sum fail due to a potential false positive?

After removing multi-sets and sets that have elements that don't exist in $S$. $S$ = $[9,6,7,4,5,1,8]$ $C$ =$[[9,6,7],[4,5],[1,8]]$ Transform the values in $C$ of the shared index values with $S$. ...
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1answer
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What practical/efficient algorithms exist for minimum weighted set cover problem (MIN-WSCP)?

For TSP there are well known heuristic and approximation solutions which run in low-polynomial time, like Christofides / 2-OPT and so on. I need a practical, fast algorithm, ideally sub-quadratic ...