Questions tagged [set-cover]

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11
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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 ...
11
votes
0answers
322 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 ...
10
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0answers
345 views

Gap hardness of Multi-Dimensional Cover

Given a finite set $X$ and a collection $F$ of subsets of $X$, we define a cover of $X$ in $F$ as a subset of $F$ whose union is equal to $X$. A cover $C$ of $X$ in $F$ is said to be exact if the ...
8
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0answers
980 views

Is Exact Cover by Equally-Sized Sets reducible to Multi-Dimensional Matching in a certain nice way?

This question is motivated by my other question “Gap hardness of Multi-Dimensional Cover,” which is in turn motivated by the question “Set Cover for Permutation Matrices” by Brayden Ware. Informal ...
7
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0answers
617 views

Hardness of Approximation results for Special Set Packing Problem Wanted

Is there any inapproximability result for the following NP-hard problem, which is a special case of the weighted Set Packing Problem? The general Set Packing Problem would be: Given A Collection of ...
5
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0answers
194 views

Can we achieve a better kernel for the Vertex Cover problem on planar graphs?

We have known how to get a $2k$ kernel for the Vertex Cover problem for thirty years, and it is not expected to be improved assuming UGC. My question is, can we do better for planar graphs? It is ...
4
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0answers
150 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 ...
4
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0answers
173 views

Approximation ratio for covering n points in d dimensions

What is best known approximation ratio for the following problem : Given n points in d dimensions , what is the minimum number of axis parallel lines needed to cover them . A line is said to cover a ...
3
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0answers
50 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 ...
3
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0answers
66 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 ...
3
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0answers
116 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 ...
3
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0answers
61 views

Do there exist “odd times” cover problems and what do we know about their approximability?

I am currently investigating a problem which can be formulated as a cover problem, in which real intervals have to be covered an odd number of times by integers. My question is just, if anybody has ...
2
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0answers
63 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,...
2
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0answers
98 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:...
2
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0answers
355 views

Why doesn't the standard analysis of set cover $H_n$ greedy extend to partial cover?

Several authors, starting with Slavik, have noted that the classical analysis of the set cover $H_n$ greedy algorithm does not readily extend to the set partial cover problem, where the goal is to ...
1
vote
0answers
204 views

Upper bound for set cover with respect to m that is better than trivial when $n \ge 3m$

Does anyone know of an upper bound for Set Cover $(\mathcal{U}, \mathcal{S}, k)$ with respect to $m=|\mathcal{S}|$ that is better than trivial when $n =|\mathcal{U}|$ is at least $3m$? (Set cover). ...
0
votes
0answers
25 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 ...
0
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0answers
184 views

Finding assignment-minimum complete k-partite graph cover

Is there any work on approximation algorithms (or exact algorithms) for finding an assignment-minimum cover of an arbitrary graph using complete k-partite subgraphs? I'm assuming this problem is NP-...
-1
votes
1answer
106 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 ...