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Questions tagged [set-cover]

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How can I optimize the assignment of object sets to workers with pre-existing caches to minimize discrepancy?

I am working on a problem where I have $n $ workers, each with a cache that already contains a specific set of objects. Additionally, I receive $n \times m$ sets of objects. My task is to assign ...
Han Tian's user avatar
1 vote
0 answers
45 views

Is the Maximum Coverage Problem Remains as Hard when Taking Most Sets?

In the maximum coverage problem (also known as max k-cover) we are given a universe $U = \{e_1,\ldots, e_n\}$ of elements, a collection $F = \{S_1,\ldots, S_m\} \subseteq 2^U$ of sets over $U$, and an ...
John's user avatar
  • 412
2 votes
1 answer
79 views

Set Cover with Multiple covers

I am interested in whether a set cover instance that covers all elements $q$ times may have the property that every sufficiently small subset of this set cover will not cover the elements even once. ...
John's user avatar
  • 412
1 vote
1 answer
116 views

Balanced set coloring

Let $\{S_1, S_2, ..., S_m\}$ be a collection of subsets of some universe $U$, where each $S_i$ has even size (so does $U$). We want to color the elements of $U$, either red or blue, such that each $...
Arnaud Casteigts's user avatar
3 votes
1 answer
140 views

Known Variant of Set Cover?

Consider the following variant of set cover: Given: Target set $T$ and a collection of sets $\mathcal{C}$, such that $T \subseteq \bigcup_{C \in \mathcal{C}} C$. Wanted: A subset $\mathcal{C'}$ of $\...
Jan Reineke's user avatar
2 votes
0 answers
42 views

Geometric Set Cover Problem and Union Complexity

I have encountered an instance of the Geometric Set Cover problem where the complexity of the union of any subset with size, say k, of m objects is linear with respect to m. I am aware of a well-known ...
Arash Vaezi's user avatar
5 votes
0 answers
62 views

W[t]-containment of partial covering problems

I would like to know more about the W[t]-containment of partial covering problems. Especially, I am interested in the question whether Partial Set Cover (Problem Definition at the end of the question) ...
xtyner's user avatar
  • 81
0 votes
1 answer
73 views

Complexity of Exact Cover problem if containing a Set Cover means there is an Exact Cover

As stated in the question, I'm interested in a variant of Exact Cover that is currently relevant to my research. Specifically, a variant where you are promised that if there is a Set Cover of size $k$,...
Matthew Ferland's user avatar
2 votes
0 answers
67 views

Confusion with the definition of Online Set Cover

I am confused on a technicality on how Online Set Cover is defined. One way to define it is: We are given a collection of sets $\mathcal{S}$ upfront, and in each time-step an element arrives to be ...
Karagounis Z's user avatar
0 votes
0 answers
129 views

Approximate inclusion-Exclusion?

I am trying to understand or find literature on the following problem of approximate inclusion exclusion. Let $S:=\{A_i\}_{i=1}^{m}$ be a set of $m$ sets. Every intersection of $k$ elements in $S$ ...
SagarM's user avatar
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1 vote
1 answer
85 views

Set cover with rewards

I am dealing with the following problem: Given a universe $U$, let $\mathcal{S}$ be a family of subsets of $U$. Each subset $S\in\mathcal{S}$ is associated with a non-negative reward, and each element ...
Mary Eiven's user avatar
0 votes
0 answers
80 views

A Simpler Solution for a special case of the Set Cover problem

We decomposed a simple polygon into many small regions. Then we estimated a visibility polygon of a point by a subset of the small regions. Now I need the minimum set of visibility polygons that can ...
Arash Vaezi's user avatar
0 votes
1 answer
63 views

Set cover where consecutive sets differ by at most one item [closed]

First I define my version of the set cover problem: We have a collection of sets such as $S_1, \dots, S_m$ where each $S_i$ is a subset of $M=\{1,\dots, m\}$. The goal is to find the minimum number of ...
Mohemnist's user avatar
  • 230
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0 answers
79 views

"Fast" approximation algorithm for geometric hitting set of same-height rectangles

In the Geometric Hitting Set problem, we are given a set of $m$ geometric objects and a set of $n$ points in $\mathbb{R}^2$, and we wish to find a small subset of the points that hits all the objects. ...
Florent Foucaud's user avatar
2 votes
1 answer
77 views

Covering a binary relation as a union of rectangles

Given finite sets $X$ and $Y$ and a subset $R\subset X\times Y$, I want to express $R$ as a union $R=\bigcup_{i=1}^n X_i\times Y_i$ with $n$ as small as possible. Here, each $X_i\subset X$ and $Y_i\...
Chris Culter's user avatar
4 votes
0 answers
48 views

Minimal partition covering?

I am working on a problem that arises in the design of experiments. I wonder if it is part of a well-studied class of problems. The problem is: Start with a set of points $S$ and a target partition of ...
CommonerG's user avatar
  • 141
-2 votes
1 answer
87 views

Set cover with small subsets [closed]

Has the variant of the Set Cover problem where each set is of size at most $d$, for some given $d$, been studied? Is it polynomial-time solvable for $d=2$ and NP-hard for $d=3$ like SAT?
user61376's user avatar
3 votes
1 answer
163 views

CNF encoding of set cover - NExpTime-completness

Notation: given a CNF formula A over variables X, we write $[A(X)]$ for the set of valuations $v: X \to \{0,1\}$ such that $A(X/v)$ is true, i.e. the set of valuations that makes formula A true. I ...
Jean-Francois Raskin's user avatar
0 votes
1 answer
519 views

What are the worst-case and average-case time complexities of the greedy algorithm for the weighted set cover problem?

Let $X$ be the universe of elements, $F$ a collection of subsets $S \subseteq X$, each with an associated cost. The goal is to find a subcollection $C \subseteq F$ of minimum total cost which covers $...
Frizzylicious's user avatar
2 votes
1 answer
108 views

Can the Banach-Tarski paradox be "realized" by floating-point round-off?

The Banach-Tarski paradox says that a ball in $\mathbb{R}^3$ can be partitioned into a finite number of pieces, whose rearrangement has a larger volume than the original. It occurred to me that it ...
Joseph O'Rourke's user avatar
3 votes
0 answers
79 views

Hardness of Approximation of Set Cover with Growing Size Bound

I'm considering the minimum set cover problem with the constraint that each set contains at most $k$ elements. Here, $k$ depends on the size of the universe. For example, $k$ may equal $\log n,\sqrt ...
Shlw Kevin's user avatar
2 votes
0 answers
77 views

Hardness result or reference for a set partition problem

I'm wondering if the following problem is (or has been proven to be) NP-Complete. Input: integer $n\ge0$, set $S_1,S_2,\ldots,S_{2n}$, set $T_1,T_2,\ldots,T_n$. Accept iff: there exists $\{a_i,...
Shlw Kevin's user avatar
6 votes
1 answer
223 views

Minimal generator for a set of sets

Is this a known problem? Given a set of sets $S$ find a set of sets $B$ s.t. each set in $S$ can be obtained through unions of some sets in $B$. The set $S$ is already a solution but the objective is ...
Mikolas's user avatar
  • 1,322
0 votes
1 answer
93 views

Polynomial approximation algorithm for set cover with assumption

We want to cover $n$ elements with some sets from $S_1, …, S_m$ (classical set cover). We furthermore suppose that any element belongs to at least $k$ sets and want to find a set cover with cardinal ...
Labo's user avatar
  • 129
2 votes
1 answer
163 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 ...
user2104150's user avatar
11 votes
0 answers
395 views

Error in paper "Some NP-complete geometric problems"?

The paper in question: M.R. Garey, R.L. Graham and D.S. Johnson. Some NP-complete geometric problems . This paper proofs the NP-completeness of some well-known problems, such as the Steiner Tree ...
J. Schmidt's user avatar
-1 votes
1 answer
87 views

Restricted Universe Exact Cover

Apologies for a simple question - I am a beginning graduate student in TCS. Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and ...
newdude's user avatar
  • 15
0 votes
1 answer
1k views

Definition of k-set cover

I'm trying to understand the sparsification lemma by Impagliazzo, Paturi and Zane (IPZ) (from this article) and in their proof they reduce the k-SAT problem to the k-set cover problem. But their ...
AstridNeu's user avatar
3 votes
1 answer
77 views

Complexity of Finding Optimal Synergistic Set Packings

Motivation: While developing tools for fast execution of machine learning workflows, we realized that many workflows require intermediate results -- sometimes we should cache these results, and ...
smacke's user avatar
  • 133
1 vote
1 answer
114 views

Set cover by alternative of sets

I had this question when looking at problem C on this morning's code jam: Suppose you have a set $S$ and $N$ pairs of subsets $\{S_i^0, S_i^1\}$, $S_i^j\subset S.$ Does there exist a cover of $S$ ...
user168715's user avatar
2 votes
0 answers
122 views

Alternative Set Cover Algorithm With Doubling

I remember that I saw once an alternative to the greedy set cover algorithm that works as follows: Assign weight 1 to every element in the universe. Repeat steps 2 and 3 until the universe is covered:...
jerry's user avatar
  • 121
4 votes
0 answers
199 views

Approximating set cover when it is known that an exact set cover exists

Suppose $U = \{1, 2, \cdots, n\}$ is a universe and $\mathcal S = \{S_1, S_2, \cdots, S_m\}$ is a collection of sets such that each set contains exactly $c$ elements, where $c$ is a constant. In this ...
taninamdar's user avatar
2 votes
1 answer
1k views

Partial cover approximation

We have a set of elements $E=\{e_1, e_2, \ldots, e_m\}$, and $n$ subsets of $E$: $S_1, S_2, \ldots, S_n$ The union of those subsets is $E$, and each subset $S_i$ has a non-negative weight $w_i$. The ...
John Harrison's user avatar
3 votes
1 answer
456 views

What is the reverse of greedy algorithm for setcover?

A common approach to approximating SETCOVER is the greedy algorithm (Algorithm 2.2 Vazirani). This algorithm greedily picks the most cost-effective subset at each iteration, removes covered elements, ...
Rahul Gopinath's user avatar
5 votes
1 answer
1k views

Is there an approximation algorithm for MAX k DOUBLE SET COVER?

Given a set system $(X,\mathcal S)$, let us say that a subset $\mathcal C\subseteq \mathcal S$ doubly covers a vertex $x$ in $X$ if $x$ is contained in at least two sets of $\mathcal C$. Let us define ...
Florent Foucaud's user avatar
2 votes
1 answer
565 views

Counting distinct set covers

I'm given a universal set $N = \{1, 2, \dots, n\}$, a family of sets $\mathcal{F} = \{ S_1, S_2, \dots, S_m \}$, $S_i \subseteq N$, and I need to count the number of distinct ways to cover the ...
roman-kashitsyn's user avatar
5 votes
1 answer
683 views

Bipartite Graphs - Maximum subset of one partition with at most n neighbours - NP-hard?

Given: A bipartite graph G=(U,V,E) Integers n and k. Decision Problem: Is there a subset of U of size k that has at most n neighbours? I am trying to figure out whether this problem is NP-hard (...
Simon Razniewski's user avatar
0 votes
0 answers
26 views

Optimizing distribution of load

I want to run $k$ programs distributed on $N$ machines. Because of resource constraints, a machine can have at most $p$ of the $k$ programs installed. To have a balanced system, I can install each of ...
Thomas Materna's user avatar
2 votes
0 answers
92 views

Variant of set cover problem with symmetric difference instead of union? [duplicate]

I am wondering if this problem has been studied, and in particular if there is an algorithm for it. Consider a universe $\mathcal U$ and a set $A \subseteq U$, and a family of sets $\mathcal F \...
Calle's user avatar
  • 121
3 votes
0 answers
73 views

Average number of sets given by greedy set cover? Is it uniform distribution?

We're covering the whole random universe $U$ of size $m$ with random sets $S_{1},\dots S_{n}$. I know that greedy set cover gives us a number between size of the minimal set cover and size of the ...
Vuk Marković's user avatar
4 votes
0 answers
173 views

Set cover approximation ratio as a function of m (number of sets)

Feige's well known result (and more recent results) show that set cover cannot be approximated within a factor of $(1 - o(1)) \ln n$, where $n$ is the number of variables. What if we want an ...
David Harris's user avatar
  • 3,508
16 votes
0 answers
501 views

a geometric variant of k-medians. NP-hard or in P?

The following problem is a special case of k-medians. Is it NP-hard? Is it in P? Input: $n$ points $(x_1,y_1), (x_2,y_2), \ldots, (x_n, y_n)$ with each $y_i \ge 0$, and an integer $k$. Output: a set ...
Neal Young's user avatar
  • 10.8k
1 vote
1 answer
106 views

Minimum order of partite in a bipartite graph

I want to create a bipartite graph where the first partite $U$ contains $L$ vertices with degree $k$ and the second partite $V$ contains $N$ other vertices with degree $a$. I need to find the minimum ...
Cantfindname's user avatar
5 votes
1 answer
323 views

Set cover in which some pairs of sets are forbidden

I'm trying to find an approximation algorithm for a variant of the weighted set cover problem. However, this variation doesn't seem to let me apply the traditional set cover arguments for an ...
Manuel Lafond's user avatar
0 votes
1 answer
56 views

1-D set cover optimisation with connected subsets

Given a 1-D universe $U$ (e.g. $\mathbb{Z}$) and a set-of-sets $\pmb{S}$, where each element of $\pmb{S}$ is a closed, connected subset of $U$ (e.g. $[a .. b]$ given $ a,b\in\mathbb{Z}$) and $\bigcup\...
John_C's user avatar
  • 103
7 votes
1 answer
683 views

Parameterized complexity of Exact Cover

Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and an integer $K$, find whether there exists a subcollection $\mathcal{S}^* \...
Piotr Skowron's user avatar
12 votes
2 answers
1k views

What is this variant of set cover problem known as?

Input is a universe $U$ and a family of subsets of $U$, say, ${\cal F} \subseteq 2^U$. We assume that the subsets in ${\cal F}$ can cover $U$, i.e., $\bigcup_{E\in {\cal F}}E=U$. An incremental ...
Alex's user avatar
  • 221
5 votes
3 answers
449 views

Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup B=...
sammy34's user avatar
  • 171
-1 votes
1 answer
128 views

Combinatorial algorithm for load balancing

I have a problem that can be solved with linear programming, but I'm hoping there's a combinatorial algorithm for this (even approximation is fine). This is basically a load balancing problem using ...
rface's user avatar
  • 1
11 votes
1 answer
951 views

Set Cover with bounded intersection size

So, the set cover problem is trivial if none of the candidate sets intersect eachother. However, what if the size of the intersection for any pair of candidate sets was at most 1? Is this problem NP-...
Garrett Andersen's user avatar