Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 35.1, Problem 3E
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To show that the professor’s heuristic does not have an approximation ratio of 2.
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Let HAM be the following special case of the Hamilton Path Problem. the Hamilton Path Problem remains NP-complete under the assumption that the graph G is planar, cubic, 3-connected, and has no face with fewer than 5 edges. (We will not need these latter two properties.)
Problem 1 A jogger wants to follow the least undesirable cycle of roads starting at her home. Each road has an "index of undesirability" and can be traversed in either direction; the jogger must follow a nonempty cycle of roads and no road can be used twice. Formulated as a graph problem, the jogger has an undirected weighted graph G = (V, E), and must determine the nonempty cycle of minimum weight starting (and ending) at vertex s.
1. Show how to use multiple applications of Dijkstra's shortest path algorithm to obtain the optimum jogger's route in time O(|V | 2 log |V | + |E||V |). Be precise: each time you want to use Dijkstra's explain which graph is the input of the algorithm,
2. Let T be the shortest path tree constructed by Dijkstra's shortest path algorithm for starting vertex s in G. Prove that some optimum jogger's route has all but one of its edges in T , and furthermore, that s is the lowest common ancestor in T of the end points of that edge.
3. Use the result in…
26. Solve the following
a) For the given set U, assume a family of subsets (F) on your own and apply the approximation algorithm to find the set cover. U={2,3,4,6,7,8}
b) For the following weighted graph, apply approximation algorithm for solving traveling salesman problem.
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27. For the following undirected graph, apply the approximation algorithm to find vertex cover.
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Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Please show written work with answer please!! A vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. Finding the minimum size vertex cover set is a NP problem, find an greedy approximation algorithm to find a close to optimum solution in polynomial time.arrow_forwardPlease solve and show all work. 21.1-2 Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V, E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V – S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V – S). Then, (u, v) is a light edge for the cut. Show that the professor’s conjecture is incorrect by giving a counterexample.arrow_forwardDemonstrate that the variant of the decision problem is NP-complete; Exists, given a graph G and a target cost c, a spanning tree in which the utmost payment of any vertex does not exceed c?arrow_forward
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- Give proof that the adapted form of the choice problem has an NP-complete solution. Is there a spanning tree in which the highest possible payment at any vertex does not exceed the target cost, given a graph G and a target cost c?arrow_forwardLet A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. What is the maximum number of edges possible for any bipartite graph between A and B?arrow_forwardGiven the adjacency matrix of an undirected simple graph G = (V, E) mapped in a natural fashion onto a mesh of size n2, in Θ(n) time a directed breadth-first spanning forest T = (V, A) can be created. As a byproduct, the undirected breadth first spanning forest edge set EA can also be created, where EA consists of the edges of A and the edges of A directed in the opposite direction.give proof of theorem.arrow_forward
- Given: graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G. Please help answer problems 3 & 4 from the given information. 3. Find an instance for which the suggested greedy algorithm gives an erroneous answer. 4. Suggest a (straightforward) algorithm which solves the problem correctly.arrow_forwardA graph G contains 20 vertices. Without using any calculator, determine themaximum number of edges the graph can have. What would be the timecomplexity of applying Bellman-Ford algorithm on the graph of n vertices.arrow_forwardLet G be a graph with n vertices. The k-coloring problem is to decide whether the vertices of G can be labeled from the set {1, 2, ..., k} such that for every edge (v,w) in the graph, the labels of v and w are different. Is the (n-4)-coloring problem in P or in NP? Give a formal proof for your answer. A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.arrow_forward
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