# 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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### Path cover with paths of bounded length, in the plane

I have a weighted, undirected, Euclidean complete graph $G$, a special vertex $r$, and an upper bound $b$. I want to find a minimum-cost path cover that covers all vertices of $G$, subject to the ...
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### Is the exact cover problem NP-hard when there is a restriction on the size?

The exact cover problem with restrictions on the size is: Input: Given a set $U=\{1,2,\ldots,n\}$ and a collection of $C$ of subsets of $U$. Question: Is there a subcollection $C^\star$ of $C$ such ...
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### What is Unique Coverage Problem?

I am trying to understand the problem statement and the approximation algorithm for the Unique Coverage Problem from these notes. I have refined the question based on the comments provided. Thanks to ...
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### Solution to a Np-hard problem and its relevance to a dual LP

From The design of APX algorithms book by David P. Williamson and David B. Shmoys, at the bottom of page 21 I saw the following statement (it is about the set cover LP and its dual): Let $y^*$ be ...
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### Problem with understanding the lower bound of OPT in Greedy Set Cover approximation algorithm

From what I know of analyzing and designing approximation algorithms, we need to find a lower bound on the optimum (in the case of minimization). For example if our solution is greedy ($SOL_G$) and if ...
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### Another version of the online set cover problem?

Here is a note about online set cover problem: we are initially given the $m$ sets, but we do not know which elements they contain. At any time $t$, we get a new element $e_t$ and learn which sets ...
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### How can I convert a list with duplicates into a set for a reduction to the set cover problem?

I'm trying to come up with a reduction for a problem whose description is more or less identical to the first problem given here. Here's a condensed version of the problem: You're given a collection ...
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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 $I = \{I_1, I_2, \ldots, I_n\}$ ...
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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 ...
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### Greedy algorithm for Set Cover problem - need help with approximation

I want to approximate how close is the greedy algorithm to the optimal solution for the Set Cover Problem, which I'm sure most of you are familiar with, but just in case, you can visit the link above. ...
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### Reduction from a further constrained problem

If I find an NP Hard problem that is equivalent to my problem with an additional constraint or bound, can I still prove that my problem is NP Hard? Generally, this is probably not the case. For ...
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 ...