Questions tagged [set-cover]

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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
36 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
54 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
310 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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170 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
81 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
65 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
63 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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65 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
173 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
37 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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2answers
731 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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1answer
10 views

Hardness of approximation statement clarification?

In the paper I'm reading, there is a hardness of approximation result for an algorithm proved using a reduction to set cover. Roughly, the claim states that if there existed an algorithm that solved ...
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35 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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62 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
205 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
145 views

Every vertex cover is a dominating set

Suppose $G$ is a connected graph and $S$ is a vertex cover. Prove that $S$ is also a dominating set. Can I get some help with proving this? I know that a dominating set in an undirected graph $G=(V,E)...
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102 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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79 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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30 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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31 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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41 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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1answer
250 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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41 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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245 views

Reduce Clique to Set Cover

Is it possible directly to reduce clique to set cover? I know that there are some ways of direct reduction from Clique to Vertex Cover and from Vertex Cover to Set Cover, so I am very interested to ...
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12 views

Set Cover with multiple covers

In the Set Cover problem we need to cover each element at least once. I'm considering the case where I want each element to be covered at least $k$ times with constant $k$. I consider the classic LP ...
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21 views

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
49 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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70 views

Variation on weighted set cover, cover by consecutive pairs

Input: A set $N = \{1, \dots, n\}$, subsets/consecutive pairs $S_1 = \{1,2\}, \dots, S_n = \{n,1\}$ with associated costs $c_1, \dots, c_n \in \mathbb{N} \cup \{0\}$, and a subset $B \subseteq N$. ...