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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.

Set cover is one of the original NP-complete problems in Karp's foundational paper. Given a collection of sets F with union U, the goal is to find a subset G⊆F, as small as possible, which also covers U. The greedy algorithm gives a logn approximation, which is worst-case optimal unless P=NP.

136 questions
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Transforming an arbitrary cover into a vertex cover

Given is a planar graph $G=(V,E)$ and let $\mathcal{G}$ denote its embedding in the plane s.t. each edge has length $1$. I have furthermore a set $C$ of points where each point $c \in C$ is contained in $\mathcal{G}$. Furthermore, it holds for any…
user695652
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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 rectangles. Is the decision version of this set cover…
Yann
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Finding a minimal cover of a subset of a finite cartesian product by cartesian products

Given a subset of a cartesian product $I \times J$ of two finite sets, I wish to find a minimal cover of it by sets which are cartesian products themselves. For example, given a product between $I=\{A,B,C\}$ and $J=\{1,2,3\}$, I may observe the…
yuvalm2
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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 shopping trip, a subset of the people travel together…
Riley
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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 many more others that can be solved by set cover.
user70945
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Problem of constructing binary sequence with least possible 1s under given constraint

You are given a binary pattern p. Problem is to construct a binary sequence of length n such that by sliding p over our sequence there is always at least one position where two 1s align (one in the pattern and other in our sequence). The goal is to…
Relja Šegvić
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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 a subcollection $S'$ of $S$ of minimum…
Manuel Lafond
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Maximum set cover with non-overlap

Let the universe be the set $U$ and a set of subsets $S$ be such that $\cup_{s \in S} s = U$. I am interested in computing the longest sequence of sets $s_1, ..., s_k$ such that: $s_i \in S$ $\forall i \in [k]$ $s_i \not\subseteq \cup_{j=1}^{i-1}…
in_question
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Algorithm to return largest subset of non-intersecting intervals

I need an efficient algorithm that takes input a collection of intervals and outputs the largest subset of non-intersecting intervals. i.e. Given a set of intervals $I = \{I_1, I_2, \ldots, I_n\}$ of the real line, we need to output a set of…
user2112791
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Proving NP-hardness of strange graph partition problem

I am trying to show the following problem is NP-hard. Inputs: Integer $e$, and connected, undirected graph $G=(V,E)$, a vertex-weighted graph Output: Partition of $G$, $G_p=(V,E_p)$ obtained by removing any $e$ edges from $E$ which…
Optimizer
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Hardness of a problem related to set cover

Suppose $C_1, \ldots, C_m$ are subsets of $\{1, \ldots, n\}$. The goal is to find the smallest subcollection of $C_1, ..., C_m$ such that each element of $\{1, \ldots, n\}$ appears at least $k$ times in this subcollection. If $k=1$, this is set…
YeLu
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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 $\mathcal{R}$ of subsets of $X$. We also have an…
Tassle
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Time Complexity of a selection problem

I wonder what's the time complexity of the following selection problem I found while thinking of a string-matching problem. [Assuming operations on integers take $O(1)$ time] We are Given $m$ sets, with $n$ integer numbers each. We want to select…
Haile
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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 whole universe: There are only ${n \choose…
Samuel Pilz
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Transition coverage for a DFA

Let $G$ be a directed graph, with a single source node $s$. I want to find a collection of paths that cover every edge of $G$ (i.e., every edge of $G$ appears in at least one of these paths), where each path must start at $s$. The cost of a…
D.W.
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