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 24.5, Problem 4E
Program Plan Intro
To prove that if sequence of relaxation steps sets
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Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.
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A Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once.
Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}.
Prove that Ham-Path is NP-hard using the reduction technique, i.e. find a known NP-completeproblem L ≤p Ham-Path (reduce from the Ham-Cycle problem). Make sure to give the followingdetails:
1) Describe an algorithm to compute a function f mapping every instance of L to an instance of Ham-Path
2) Prove that x ∈ L if and only if f(x) ∈ Ham-Path
3) Show that f runs in polynomial time
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- A Hamiltonian path on a directed graph G = (V, E) is a path that visits each vertex in V exactly once. Consider the following variants on Hamiltonian path: (a) Give a polynomial-time algorithm to determine whether a directed graph G contains either a cycle or a Hamiltonian path (or both). Given a directed graph G, your algorithm should return true when a cycle or a Hamiltonian path or both and returns false otherwise. (b) Show that it is NP-hard to decide whether a directed graph G’ contains both a cycle and a Hamiltonian Path, by giving a reduction from the HAMILTONIAN PATH problem: given a graph G, decide whether it has a Hamiltonian path. (Recall that the HAMILTONIAN PATH problem is NP-complete.)arrow_forwardProve that any graph with the degree sequence 2,5, 5, 7, 7, 7,7, 7, 7 is Hamiltonian. A complete tripartite graph is formed by taking three groups of vertices A, B, and C, then adding an edge between every pair of vertices in different groups. We write Kabe for the complete tripartite graph with |A| = a, |B| = b, and |C| = c. For example, below are diagrams of K2,4,5 (left) and K23,6 (right). (a) One of these graphs is Hamiltonian. Find a Hamiltonian cycle in that graph. (b) The other of these graphs is not Hamiltonian. Give a reason why it does not have a Hamiltonian cycle. (c) Generalize: find and prove a rule that tells you when Kabe is Hamiltonian. You may assume aarrow_forwardAn independent set of a graph G = (V, E) is a subset V’ Í V of vertices such that each edge in E is incident on at most one vertex in V’. The independent-set problem is to find a maximum-size independent set in G . Analyse the complexity of the approximation algorithm for solving the independent-set problemarrow_forwardIf a graph has a collection of subsets of the edge set E, with the edges of at most one cycle: Show that if |X| and |Y| are independent sets, show that |X| < |Y| implies that there exists {m} E Y\X such that X U {m} is independent.arrow_forwardShow that if all edges of a graph G have pairwise distinct weights, then thereis exactly one MST for G.arrow_forwardGiven a graph G = (V, E) find whether there exists an independent set of size 8. Is the problem N-P complete or not. Why?arrow_forwardAssume that the Hamiltonian cycle problem for undirected graphs is NP-complete. Show that the Hamiltonian cycle problem is NP-complete for directed graphs. Note: Just need to reduce the Hamiltonian cycle problem for an undirected graph to a directed graph: for any edge (u, v) in the undirected graph, add two directed edges u->v and v->u.arrow_forwardPropose an approximation algorithm for solving the independent-setproblem where an independent set of a graph G = (V, E) is a subset V’ is a subset of V of verticessuch that each edge in E is incident on at most one vertex in V’. Theindependent-set problem is to find a maximum-size independent set in G.arrow_forwardShow that a bottleneck SPT of a graph is identical to an MST of an undirected graph. It provides the path between each pair of vertices v and w whose longest edge is as short as feasible.arrow_forward2. Prove that in a graph G, Sc V(G) is an independent set if and only if S is a vertex cover and hence a(G)+B(G) = n(G).arrow_forwardMy professor went over the proof on this slide and I don't really understand it. Can you explain it in detail step by step?arrow_forwardProve that if an edge (u, v) is in a MST for a graph G, then (u, v) is a light edge crossing some cut in G.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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