Homework Answers
Solution:
The given problem is NP, because in polynomial time it is able to verify that there is a subgraph with k nodes atleast k weight.
This is NP complete problem. It can be reduced from k-clique which is NP complete and checking the total weight is atleast α
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NP-completeness. We are given an undirected graph where each edge has a positive weight. Given (k,...
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph...
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph G (V, E), each edgee which can be positive or negative. The zero weight cycle problem is that whether exists a simple cycle (each vertex passes at most once) to make the sum of the weights of each edge in G is exactly equal to 0. Prove that the problem is NP complete.
3, (30 points) Given...
We now consider undirected graphs. Recall that such a graph is • connected iff for all...
We now consider undirected graphs. Recall that such a graph is • connected iff for all pairs of nodes u, w, there is a path of edges between u and w; • acyclic iff for all pairs of nodes u, w, whenever there is an edge between u and w then there is no path Given an acyclic undirected graph G with n nodes (where n ≥ 1) and a edges, your task is to prove that a ≤ n...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete.
Note:...
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.
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...
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph...
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph induced on S, denoted G[S] is a graph that has vertex set S and an edge between two vertices u, v that is an element of S provided that {u,v} is an edge of G. A subset K of V is called a killer set of G if the deletion of K kills all the edges of G, that is...
Problem 3: Suppose you are given an undirected graph G and a specified starting node s...
Problem 3: Suppose you are given an undirected graph G and a specified starting node s and ending node t. The HaMILTONIAN PATH problem asks whether G contains a path beginning at s and ending at t that touches every node exactly once. The HAMILTONIAN CYCLE problem asks whether con- tains a cycle that touches every node exactly once (cycles don't have starting or ending points, so s and t are not used here) Assume that HaMIlTonian CYCLe is NP-Complete....
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G wi...
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G with positive integer distances on the edges, and two integers f and d, is there a way to select f vertices on G on which to locate firehouses, so that no vertex of G is at distance more than d from a firehouse?
Problem 6. In lecture, we saw that an undirected graph with n nodes can have at...
Problem 6. In lecture, we saw that an undirected graph with n nodes can have at most n(n - 1)/2 edges. Such a graph necessarily has one connected component. The greatest number of edges possible in a disconnected graph, however, is smaller. Suppose that G (V, E) is a disconnected graph with n nodes, how large can |El possibly be? You do not need to prove your answer, but you should provide some explanation of how you obtained it.
Question 6 15pt Given undirected positive edge weight graph G(V.E) and boo(m) = m3 while (v...
Question 6 15pt Given undirected positive edge weight graph G(V.E) and boo(m) = m3 while (v € VX for all u € Adj[v]) { boo(V): ) ) What is the run-time of the code above given the Graph is store as adj. list? What is the run-time of the code above given the Graph is store as adj. matrix? Please explain your answer.
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph G (V, E), each edgee which can be positive or negative. The zero weight cycle problem is that whether exists a simple cycle (each vertex passes at most once) to make the sum of the weights of each edge in G is exactly equal to 0. Prove that the problem is NP complete.
3, (30 points) Given...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete.
Note:...
Problem 3: Suppose you are given an undirected graph G and a specified starting node s and ending node t. The HaMILTONIAN PATH problem asks whether G contains a path beginning at s and ending at t that touches every node exactly once. The HAMILTONIAN CYCLE problem asks whether con- tains a cycle that touches every node exactly once (cycles don't have starting or ending points, so s and t are not used here) Assume that HaMIlTonian CYCLe is NP-Complete....
Problem 6. In lecture, we saw that an undirected graph with n nodes can have at most n(n - 1)/2 edges. Such a graph necessarily has one connected component. The greatest number of edges possible in a disconnected graph, however, is smaller. Suppose that G (V, E) is a disconnected graph with n nodes, how large can |El possibly be? You do not need to prove your answer, but you should provide some explanation of how you obtained it.
Question 6 15pt Given undirected positive edge weight graph G(V.E) and boo(m) = m3 while (v € VX for all u € Adj[v]) { boo(V): ) ) What is the run-time of the code above given the Graph is store as adj. list? What is the run-time of the code above given the Graph is store as adj. matrix? Please explain your answer.
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