Questions tagged [independent-sets]
15 questions
3
votes
1 answer
Prove that "max independent set is larger than max clique" is NP-Hard
We define B as:
$B = \{ | \text{ G is an undirected graph in which} \\
\text{the number of vertices in the largest independent set} \\
\text{is greater than the number of vertices in the largest clique} \}$
Can you prove that B is NP-Hard? I…
Shlomiz
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3
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Minimum number of edges required to limit the size of the independent set
Consider the following problem:
For $k \le n$, what is the smallest number of edges in the graph with $n$ vertices so that the maximum independent set has size at most $k$?
I.e., given an empty graph on $n$ vertices, I want to add the smallest…
Dmitry
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3
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1 answer
How can I model this optimization problem?
We're looking to model the following problem as a standard optimization problem (or even a non-standard one). We can come close, but nothing seems to fit exactly. We have a working algorithm coded, but the performance is unacceptable.
Problem…
Ted Hopp
- 133
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3
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1 answer
Maximum Independent Set of a Tree using Greedy Algorithm
I was attempting to solve "Maximum Independent Set of a Tree" and came up with an algorithm that resembled this one Why is greedy algorithm not finding maximum independent set of a graph?
Code extract shown here:
Greedy(G):
S = {}
While G is not…
wamengti
- 31
- 2
2
votes
1 answer
How to express that a graph has Independent set of size at least $n/2$ in $\exists SO$
I am looking to provide a formula saying "A graph with $n$ vertices has an independent set $X$ of size at least $n/2$" in existentional second order logic.
(This is exercise 1.2. from Libkin's Elements of Finite Model Theory (see here for online…
Michal Dvořák
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2
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1 answer
Why is Independent Set "at least" and Vertex Cover "at most" k
The decision version of the Independent Set and Vertex Cover problems are phrased as:
Given a graph G and a number k, does G contain an independent set of size at least k?
Given a graph G and a number k, does G contain a vertex cover of size at…
nicetyartwork
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1
vote
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Inapproximability of Maximum Independent Set
Lemma1. There exists a polynomial time computable transformation $f$ from the 3CNF formulas to graphs such that for every 3CNF formula $\varphi$, $f(\varphi)$ is an $n$-vertex graph whose $|MIS| = \mathcal{V}(\varphi).\frac{n}{3}.$
(We denote…
Monte_carlo
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1
vote
1 answer
$f$ is Reduction from $\texttt{INDSET}$ to itself
My teacher said in his lecture( followed by the book Barak and Arora) the following:
We will imagine that a shocking discovery reveals that there exists a function $f$, thinking in linear time, so that for every input of the form $\langle G,…
user150715
1
vote
1 answer
Reducing the Independent Set Problem to Independent Set for 3-Colorable Graphs
I am exploring a reduction from the general Independent Set Problem to the Independent Set Problem specifically for 3-colorable graphs. The goal is to demonstrate that the maximal independent set of a general graph can be determined using an…
Ferran Gonzalez
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1
vote
1 answer
Independent sets generation in a graph
Is there an algorithm that, given an undirected graph and one independent set IS1, finds an other independent set IS2 by adding and deleting vertices from the first IS1?
maliya
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1
vote
1 answer
Densest Sub Graph and forbidden Pairs
Given two graphs $G$ and $F$ on the same vertex set $V$. Compute a sub set $\tilde{V}\subset V$ which' sub graph of $G$ is of maximum density and does not have any pair that is connected in $F$.
Formally, find
$$
arg\max_{\tilde{V}\subset…
Daniel Schwegler
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1
vote
1 answer
Prove the expected size of the independence set got by a random algorithm is at least 1/d of the maximum size
I am doing an exercise related to maximizing Independent Set, I have $G = (V = \{v_1, . . . , v_n\}, E)$ as an undirected graph. This graph as $n!$ possible orderings for the vertices $V$.
If we pick any such orderings uniformly at random i.e. let…
ConScience
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0
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Reduction from satisfiability to independent set in maximum variation
Main question is: I want to do reduction:
$$\text{MAX-3SAT}\leq_p\text{MAX-INDSET}.$$
My tried so far:
But I know the reduction $\text{3SAT}\leq_p\text{Independent Set}.$
Input: Given a 3CNF formula $\varphi$.
Goal: Construct a graph $G_\varphi$…
Monte_carlo
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0
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Find an independent set in which the cumulative sum of weights is maximized
I have a weighted undirected graph G=(V,E,W), I want to find an independent set S of V, such that if we sort the vertices of S in increasing order of their weights, the cumulative sum of the weights is maximized. In other words, if we suppose…
Farah Mind
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0
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Is maximal independent set on maximal planar graphs still NP-complete?
We know that finding the size of the maximal independent set of a planar graph is NP-complete. I'm curious about whether it remains NP-complete for maximal planar graphs, i.e., the graphs in which adding any edge will destroy its planarity, or…
Soha
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