Homework Answers
If x1 = True, x2 = False, x3 = True, x4 = True then:
f = 
Thus, f is satisfiable.
So, we must construct a graph G which has a clique of size 3, because then only the "if and only if"statement would be true.
Note that if f hasn't been satisfiable then we would rather have drawn graph G which didn't have a clique of size 3, because then only the "iff" statement have been satisfiable.
Thus, our G can be simply a triangle then, because it's a clique of size three.
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Problem . Given the formula f- construct a graph G such that f is satisfiable iff G has a clique ...
2. For a given graph G, we say that H is a clique if H is...
2. For a given graph G, we say that H is a clique if H is a complete subgraph of Design an algorithm such that if given a graph G and an integer k as input, determines whether or not G has a clique with k vertices in polynomial time. (Hint: Try to first find a polynomial time algorithm for a different problem and reduce the clique problem to that problem).
2. For a given graph G, we say that H is a clique if H is a complete subgraph of Design an algori...
2. For a given graph G, we say that H is a clique if H is a complete subgraph of Design an algorithm such that if given a graph G and an integer k as input, determines whether or not G has a clique with k vertices in polynomial time. (Hint: Try to first find a polynomial time algorithm for a different problem and reduce the clique problem to that problem).
2. For a given graph G, we say that...
1) Consider the clique problem: given a graph G (V, E) and a positive integer k, determine whethe...
1) Consider the clique problem: given a graph G (V, E) and a positive integer k, determine whether the graph contains a clique of size k, i.e., a set of k vertices S of V such that each pair of vertices of S are neighbours to each other. Design an exhaustive-search algorithm for this problem. Compute also the time complexity of your algorithm.
4. Approximating Clique. The Maximum Clique problem is to compute a clique (i.e., a complete subgraph) of maximum size i...
4. Approximating Clique. The Maximum Clique problem is to compute a clique (i.e., a complete subgraph) of maximum size in a given undirected graph G. Let G = (V,E) be an undirected graph. For any integer k ≥ 1, define G(k) to be the undirected graph (V (k), E(k)), where V (k) is the set of all ordered k-tuples of vertices from V , and E(k) is defined so that (v1,v2,...,vk) is adjacent to (w1,w2,...,wk) if and only if, for...
Part A: [5 pts] Show that CLIQUE-COVER ∈ NP NP Completeness Proof The CLIQUE-COVER problem is...
Part A: [5 pts] Show that CLIQUE-COVER ∈ NP NP Completeness Proof The CLIQUE-COVER problem is defined as follows: Given a graph G, which has a number of cliques ci, c2, …, cm (m ≥ k), and a number k, the CLIQUE-COVER problem is the problem of determining whether all the nodes of the graph are covered by (i.e., contained in) at most k of the cliques of nodes. See Appendix 2 for an example of a graph with a...
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤...
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤ n), the CLIQUE problem asks us whether there is a set of k vertices in G that are all connected to one another. That is, each vertex in the ”clique” is connected to the other k − 1 vertices in the clique; this set of vertices is referred to as a ”k-clique.” Show that this problem is in class NP (verifiable in polynomial time)...
Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and...
Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and an integer k, return a clique of size k as well as an independent set of size k, provided both exist.
4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there...
4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there a truth assignment for the variables such that each clause has at least one true literal and at least one false literal? The NOT-ALL-EQUAL 3SAT problem is NP-complete. This question is about trying to reduce the NOT-ALL-EQUAL 3SAT problem to the MAX-CUT problem defined below to show the latter to be NP-complete. A cut in an undirected graph G=(V.E) is a partitioning of the...
Algorithms Given the following 3SAT formula, convert the problem to Independent Set and determine from there...
Algorithms
Given the following 3SAT formula, convert the problem to Independent Set and determine from there if the formula is satisfiable: $ = (x1 V x2 V x3) ^ (X1 V X2 V x3) ^ (X1 V x2 V x3)
Definition: Given a Graph \(\mathrm{G}=(\mathrm{V}, \mathrm{E})\), define the complement graph of \(\mathrm{G}, \overline{\boldsymbol{G}}\), to be \(\bar{G}=(\mathrm{V},...
Definition: Given a Graph \(\mathrm{G}=(\mathrm{V}, \mathrm{E})\), define the complement graph of \(\mathrm{G}, \overline{\boldsymbol{G}}\), to be \(\bar{G}=(\mathrm{V}, E)\) where \(E\) is the complement set of edges. That is \((\mathrm{v}, \mathrm{w})\) is in \(E\) if and only if \((\mathrm{v}, \mathrm{w}) \notin \mathrm{E}\) Theorem: Given \(\mathrm{G}\), the complement graph of \(\mathrm{G}, \bar{G}\) can be constructed in polynomial time. Proof: To construct \(G\), construct a copy of \(\mathrm{V}\) (linear time) and then construct \(E\) by a) constructing all possible edges of between vertices in...
2. For a given graph G, we say that H is a clique if H is a complete subgraph of Design an algorithm such that if given a graph G and an integer k as input, determines whether or not G has a clique with k vertices in polynomial time. (Hint: Try to first find a polynomial time algorithm for a different problem and reduce the clique problem to that problem).
2. For a given graph G, we say that H is a clique if H is a complete subgraph of Design an algorithm such that if given a graph G and an integer k as input, determines whether or not G has a clique with k vertices in polynomial time. (Hint: Try to first find a polynomial time algorithm for a different problem and reduce the clique problem to that problem).
2. For a given graph G, we say that...
4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there a truth assignment for the variables such that each clause has at least one true literal and at least one false literal? The NOT-ALL-EQUAL 3SAT problem is NP-complete. This question is about trying to reduce the NOT-ALL-EQUAL 3SAT problem to the MAX-CUT problem defined below to show the latter to be NP-complete. A cut in an undirected graph G=(V.E) is a partitioning of the...
Algorithms
Given the following 3SAT formula, convert the problem to Independent Set and determine from there if the formula is satisfiable: $ = (x1 V x2 V x3) ^ (X1 V X2 V x3) ^ (X1 V x2 V x3)
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