# Questions tagged [set-cover]

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### 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 ...
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 ...
1answer
33 views

### Is this variation of set-cover NP-hard to approximate?

The classic set-cover problem is described as follows: Let $S = \{s_1, ..., s_n\}$ be a target set, and let $\Lambda = \{A_1, ..., A_m: A_i \subset S\}$ be a collection of subsets of $S$. The ...
1answer
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### 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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### Specialized Algorithm for Set-Cover with $k=3$

I know that the Set-cover problem with $n$ elements and a universe of size $N$ is NP complete. Also, the problem is has parameterized complexity regarding the number of sets $k$ that should cover the ...
1answer
43 views

### Probability of randomly designated subsets cover the universe

Let $U=\{1,2,\ldots,n\}$ and $S \subseteq \mathscr{P}(U)$. Let $T$ be a subset of $S$, randomly constructed selecting independently each element of $S$ with probability $p$. Is there a polynomial ...
1answer
33 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 ...
1answer
39 views

### How hard is it to decide if there exists a strict improvement of a given solution of an NP-complete problem?

Take the Set Cover problem as an example. When we ask if there is a set of size k that covers all the elements, the problem is NP-complete. Now if we ask, for a given set $S$ of size $k$, if there ...
1answer
144 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 ...
1answer
23 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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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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76 views

1answer
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### 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 ...
3answers
1k views

### What are the real world applications of set cover problem?

I am studying about set cover problem and wondering that which problems in real world can be solved by set cover. I found that IBM used this problem for their anti-virus problem, so there should be ...
0answers
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### 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 ...
0answers
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\...
1answer
169 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 ...
2answers
730 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 ...
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 ...
0answers
50 views

### 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 ...
1answer
431 views

### Grid covering by rectangles

We have a $N_1 \times N_2$ grid. We have a collection of rectangles on this grid, each rectangle can be represented as a $N_1$-by-$N_2$ binary matrix $R$. We want to cover the grid with those ...
1answer
404 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? ...
0answers
127 views

### 2-approximation edge-cover algorithm using primal-dual method

The problem Given an undirected graph $G=\left(V, E\right)$ and positive edge weights $w_e$, design a 2-approximation algorithm based on the primal-dual principle. So far I managed to represent the ...
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34 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 ...
1answer
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### 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 ...
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, ...
0answers
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?
1answer
244 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 ...
0answers
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 $B \subseteq N$. ...
1answer
145 views

### Is 'double max-$k$ vertex-cover' NP-hard?

Consider the following problem, which I have called '$\mathsf{\text{double max$k$-vertex-cover}}$': Given an undirected graph $G=(V,E)$ and integers $k$ and $t$, does there exist a set of vertices ...
2answers
2k 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 ...
4answers
811 views

### Minimum number of shopping trips for a group of people to buy presents for each other

We have a group of $n$ people. We are given a list of who must buy presents for whom within the group. Each person might need to buy/receive any number of presents, or possibly none at all. In a ...
1answer
196 views

### Is this variant of set cover NP-complete?

Let's say we have a universe $U$ of $n$ elements and a collection $S$ of $m$ subsets of $U$, i.e., $S=\{S_1,\ldots,S_m\}$, and a positive integer $k$. If I ask "is there a set cover of $U$ of size $k$ ...
1answer
230 views

### Approaches to the size constrained weighted set cover problem

I am trying to solve a weighted set cover problem where the number of selected subsets is limited by a constant $k$. Assuming this is a pretty straight-forward variation of weighted set cover I ended ...
1answer
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### 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 ...
1answer
56 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 \$\...