程序代写 Problem Hamiltonian Cycle – cscodehelp代写
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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