Problem Hamiltonian Cycle

A cycle C in

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if itvisitseach

is Hamiltonian Cycle

vertex exactly once Problem statement

Given an undirectedgraph G is there a Hamiltonian cycle in G

Show that the Hamiltonian Cycle Problem is NP complete

A suppose that G

a vertex cover ofSgi E

vertex cover setbe

Cv E has let the

we will identify neighbors of Ui as shown here

gg by following

Form a Ham Cycle in G

the nodes in start at s

u viif u iUfif

G in this order i

Then go to Sz and follow the nodes

UE I yds if

fun under D

Ye Thenreturnbackto s

Suppose G has a Hamiltonian cycle C then the set

S uj ell i ui D e C

for some k j will be a vortex cover set in G

Theorem if P NP then for any constant971 there is

no polynomial time approximation algorithm with approximation

ratio f for the general TSP Plan we will assume thatsuch

approximation algorithm

exists we will then use solve the H C problem

Given problem

construct G as

an instance ofthe on graph G we

follows has the same setnodes

as in G connected

G is a fully have a cost of 1

graph Edges in G that are also G

otheredges in 0 have a

Discussion 11

1. In the Min-Cost Fast Path problem, we are given a directed graph G=(V,E) along with positive integer times te and positive costs ce on each edge. The goal is to determine if there is a path P from s to t such that the total time on the path is at most T and the total cost is at most C (both T and C are parameters to the problem). Prove that this problem is NP-complete.

2. We saw in lecture that finding a Hamiltonian Cycle in a graph is NP-complete. Show that finding a Hamiltonian Path — return to its starting point — is also NP-complete.

3. Some NP-complete problems are polynomial-time solvable on special types of graphs, such as bipartite graphs. Others are still NP-complete.

Show that the problem of finding a Hamiltonian Cycle in a bipartite graph is still NP-complete.

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