Questions tagged [vertex-cover]
91 questions
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Compute the expected size of an approximation of vertex cover
Consider the following randomized approximation algorithm of vertex cover:
Input: A graph G = (V, E).
Output: A set $C_G \subseteq V$ a vertex cover of $G$.
The algorithm:
Set $C_G := \emptyset$.
For each edge $e$, in some arbitrary order:
If…
Narek Bojikian
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5
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A problem with the greedy approach to finding a maximal matching
Suppose I have an undirected graph with four vertices $a,b,c,d$ which are connected as in a simple path from $a$ to $d$, i.e. the edge set $\{(a,b), (b,c), (c,d)\}$. Then I have seen the following proposed as a greedy algorithm to find a maximal…
IntegrateThis
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4
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1 answer
Vertex cover with covering radius 2
Our problem is: Given an undirected graph, does it have a vertex cover consisting of $k$ vertices?
A vertex included in this vertex cover variant will cover every edge incident to it and every edge incident to some edge incident to it. Hence, we…
Thinh D. Nguyen
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4
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3 answers
Reduce Vertex Cover with size k to Vertex Cover with size n/2
Disclaimer: This is a homework question.
I would like to reduce vertex cover problem to the following problem:
$$L = \{G \mid G\text{ has a vertex cover of size } |V(G)|/2\}\,.$$
I have divided the problem into three parts:
$k = n/2$: This case is…
Henry Cho
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4
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Smallest unions of shifts to cover the entire set?
Consider a subset $A\subseteq \{0, 1, 2, \dots, n - 1\}$. We define $A + i = \{(a + i) \text{ mod } n \mid a \in A\}$. For example if $A = \{1, 3, 4\}$ and $n = 5$, $A + 2 = \{3, 0, 1\}$.
We want to find the smallest number of these shifted subsets…
Maximilian Meiler
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4
votes
2 answers
Vertex cover approximation: what's wrong with max-degree heuristic?
For context: the usual greedy approximation algorithm for the minimum vertex cover problem (given a graph, find the smallest set of vertices such that every edge is incident to at least one selected vertex) goes like this:
while some uncovered edge…
Kye W Shi
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4
votes
1 answer
FPT algorithm for 1-BDD
Given a graph $G = (V,E)$ and an integer $k$, the 1-BDD problem asks if there exists a subset $D$ of at
most $k$ vertices such that the degree of any vertex in $G[V \setminus D]$ is at most one.
Is there any FPT algorithm for the above problem…
Jumanji Halastra
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4
votes
1 answer
Finding a kernel for d-Bounded degree deletion
In $d$ Bounded degree deletion problem, we are given an undirected graph $G$ and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the the maximum vertex degree of the graph to at most $d$.
The…
advocateofnone
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4
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How to cover given entries in a matrix with minimum number of rows and columns?
We have a matrix of 0 and 1. We want to cover all the 1's. We can cover a row or a column with a plate. We want to use the minimum number of plates.
Example:
\begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1 &…
user102578
4
votes
1 answer
Vertex Cover of size at most $\log n$
Consider the following language:
$$
L = \{ G | G \text{ has a VC of size at most } \log n \}
$$
Does $L\in P$ or $L\in NPC$?
galah92
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4
votes
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Does the intersection of VC and CLIQUE belong to NPC?
Define: $$L=\{(G,k) : G\text{ has a vertex cover of size at most $k$, and a clique of size at least $k$}\}$$
I need to determine whether $L\in \mathrm{NPC}$ or $L\in \mathrm{P}$. I suspect that $L\in \mathrm{NPC}$, been trying to prove it by…
John
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3
votes
1 answer
Vertex Cover on Comparability Graphs
Is there anything known about the hardness of Vertex Cover on the subclass of comparability graphs? In particular, is it known whether the problem is still NP-hard?
Related Results: In "Modular decomposition and transitive orientation, McConnell &…
Henning Koehler
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3
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Minimum Vertex Cover of 2 vertex disjoint odd cycles that have edges between them
Consider the graph $G$, which is comprised of 2 vertex disjoint odd cycles ($C_1$, $C_2$) where $|C_1|$ and $|C_2| \geq 5$. $G$ is sub-cubic and connected, with edges in between the cycles. Because $G$ is sub-cubic, each node's degree $\leq 3$ but…
Dabbler
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3-cycle cover decision problem for directed graphs: best known algorithm and maximum size of tractable problems
I know that the 3-cycle cover decision problem for directed graphs (3-DCC), defined as finding whether a directed graph has a disjoint vertex cycle cover in which every cycle has at least 3 edges, is NP-complete.
Does anyone know what is the actual…
Eugenio
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3
votes
1 answer
A question about the work per recursive call in FPT vertex cover of size k algorithm
I have been looking at the FPT(Fixed Parameter) algorithm for checking if a vertex cover of size k exists.The algorithm goes as follows:
VertexCoverFPT$(G, k)$
if $G$ has no edges then return true
if $k=0$ then return false
let $uw$ be some edge of…
temp3318
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