Questions tagged [set-cover]

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minimum cover intervals in a subset

I've been going about this problem for day and can't seem to understand how will I solve this? Canon is faced with a set $A$ of $n$ intervals on the real line. He is looking for a subset of these ...
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36 views

Minimum Dominating Set

Consider a graph $G$ with minimum degree $d$, we know through sets cover, it's possible to find the one dominating set $S$ that covers $G$ such that $$S\leq O(\log n)\frac{n}{d} $$ with high ...
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35 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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1answer
93 views

Is this a known problem in graph theory?

My basic problem includes a graph where each node $i$ is associated with a weight $c_i$, and the problem is to find a minimum (or maximum) weighted independent set with a fixed cardinality $p$. This ...
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1answer
45 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
23 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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19 views

Set Cover: Understanding the algorithm with an example

I am trying to follow the following link: Solution for an example They have provided solution for an example using the greedy algorithm. I have got following questions: (1)Why start with Z, cost is 7, ...
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1answer
65 views

Maximum coverage 1/2-approximation algorithm: why does the central lemma hold?

I am looking for an approximation algorithm for the Maximum Coverage problem and a proof of its approximation ratio. As approximation algorithm I use the greedy algorithm which chooses the set that ...
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1answer
16 views

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 ...
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1answer
485 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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22 views

Set cover problem satisfying half constraints

Consider a variation of the set cover problem in which we only have to cover some certain amount of elements. Is this problem still NP-hard? I guess yes? Is there a good reference?
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1answer
83 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
67 views

Upper bound for size of minimal covers of a set

Would appreciate any insight about the following regarding set covers: Begin with a universe set $X = \{x_1,x_2,...,x_n\}$ and a set $S=\{s_1, s_2,...,s_p\}$ such that each $s_i \subseteq X$ and $\...
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40 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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1answer
144 views

maximum coverage version of dominating set

The dominating set problem is : Given an $n$ vertex graph $G=(V,E)$, find a set $S(\subseteq V)$ such that $|N[S]|$ is exactly $n$, where $$N[S] := \{x~ | \text{ either $x$ or a neighbor of $x$ ...
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79 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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1answer
303 views

Hitting Set Problem with non-minimal Greedy Algorithm

The Hitting Set Problem is defined as having a universal set $\mathfrak{U}$, and nonempty sets $S_i \subseteq \mathfrak{U}$ for $1 \leq i \leq n$, and finding a set $\mathcal{H} \subset \mathfrak{U}$ ...
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1answer
290 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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35 views

Point of Interest Problem

You are in a xy plane with a set of points F. You also have a collection P of N sets { P1,...., Pn} where each of the set consist of points of the form (Px,Py). Each set has a different number of ...
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1answer
92 views

Minimal hitting set with respect to set inclusion from a book “Parameterized Complexity Theory”

In the first chapter of "Parameterized Complexity Theory" by Flum and Grohe, an example is presented to find a hitting set of minimal cardinality. In Fig. 1.3, the author says a black colored leaf ...
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20 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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1answer
31 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 ...
3
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1answer
66 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 ...
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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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42 views

Weighted Set Cover Problem Minimizing Average Weight

In the traditional weighted set cover problem, we aim at minimizing the sum of the weight of the selected sets. Is there any problem/literature that aim at minimizing the average weight of the ...
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1answer
62 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
171 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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116 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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1answer
84 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
104 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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2answers
545 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
1k views

Given a set of intervals on the real line, find a minimum set of points that “cover” all the intervals

I've been trying to find an efficient way to solve the problem of finding a minimum (not minimal) set of time points that cover a given family of intervals on the real line, that is, for each interval ...
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1answer
1k 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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69 views

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 ...
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1answer
49 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 ...
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1answer
43 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
51 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 ...
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1answer
395 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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1answer
26 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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140 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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99 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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1answer
161 views

An exact solution for biclique vertex-cover problem on a bipartite graph

The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most "k" bicliques (complete bipartite subgraphs). It has been shown that "Biclique Vertex-...
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1answer
69 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
514 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
371 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
137 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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39 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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34 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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1answer
213 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
104 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 ...