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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Exact Cover variant: partition a family of subsets into exact coverings

I have found that a problem that I'm analyzing is equivalent to the following variant of the Exact Cover problem: Partition into $k$ Exact Covers Input: A universe ...
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Is Set Cover problem with subsets of size ≤2 solvable in polynomial time?

I came across the below question where the polynomial time solution to the "Set Cover Problem" is discussed when the subsets are of size EXACTLY 2. Set cover problem with sets of size 2 The ...
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Trivial vertex cover in regular graph is 2-approximation Proof

I need to show that in any regular graph, taking all nodes gives a 2-approximation vertex cover. My attempt: I am proving that every $k$-regular graph can be reduced to a 2-regular fully connected ...
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Prove that a dominating set has minimum cardinality in a "unit interval graph"

I am given the definition of a unit interval graph, e.g. $G = (V, E)$ such that $\forall v \in V$ there is a weight $x_v \in \mathbb{R}$ and nodes $u, w$ has an edge iff $|x_u - x_w| < 1$. I am ...
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Solution methods for this Weighted Partial Set Cover-ish problem

Given a set of subsets $S_1, ..., S_N$ of a finite universe $E$ of elements $e_1, ..., e_n$ and mapping of those elements to an integer 'weight' $w_1, ... w_n$, select the subset of subsets which ...
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What is the name for this minimal satisfying set covering problem? [duplicate]

Preface Hello! I have a problem here that's difficult for me to Google, and I don't know if there's a name for it. It feels like a set cover problem of some kind, but I'm very unfamiliar with ...
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How to "sort" stops when combining multiple bus routes (joint timetable)?

In a public transport system, it can happen that trips with partially different routes are written on one timetable. For example, if TRIP 1 has stops A-B-C-D and TRIP 3 has stops A-B-C-E, then the ...
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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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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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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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Solving Budgeted Maximum Coverage Problem using Greedy and Genetic Algorithm

I am trying to solve the Budgeted Maximum Coverage Problem. I have read and implemented the greedy and modified-greedy methods to solve it, as proposed by Khuller. Both are approximation algorithms. ...
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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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Minimum set cover with incompatible sets

I'm interested in a variant of minimum set cover where some sets are ``incompatible'' (they can't be chosen simultaneously). To state it more formally: We have a finite base set $X$ and a family $\...
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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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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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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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Approximation of Set Cover

I wonder why do we say $\log n$ is the best possible approximation factor for Set Cover Algorithm? We already know there exists a 2-approximation algorithm for vertex cover, which is obviously better ...
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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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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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1 answer
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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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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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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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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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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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1 answer
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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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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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3 votes
1 answer
105 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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2 votes
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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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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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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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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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1 answer
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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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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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3 votes
1 answer
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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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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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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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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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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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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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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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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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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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1 answer
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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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1 answer
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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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1 vote
2 answers
756 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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1 answer
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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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1 answer
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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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5 votes
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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 ...
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