Introduction to Algorithms
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 s.π to a non-NIL value, then G contains a negative-weight cycle.

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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.
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
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 belongs to NP, i.e. given any graph G and a pair of nodes u and v and apath p between u and v, there is a polynomial algorithm to check whether p is in Ham-Path
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