a Certificate r
A tour ofcost at mostD
b Catterall
checks we didforftc
check that total costof
the four E D
claim G has a H C iff
we have a tour ofCort E n is G
a if There is a HC in G
Tour of art n in G
b if we have a Tourofcostnis G There will be
a HC is a
NP Completeproblems we knowof
3 SAT in deep set
Given a setof a items where
i
Decision version of
oil knapsack
as
hH
Cotidour 2 Castoff
costof initial tour 2 artyMST
coatof our approxSol Ez CostofMST
costof our approx Sol s centfoptitoon
This is a E approximation algorithm
General TSP
Theorem if P NP then for
any constant971 there isno polynomial time approximation
algorithm with approximation
ratio f for the general TSP
Plan we will assume thatsuch
an approximation algorithm
exists we will then use it to
solve the H C problem
Given an instanceofthe H C
problem on graph G we
will construct G as follows
o G has the same setnodes
as in G
o G is a fullyconnectedgraph
Edges in G that are also G
have a cost of 1otheredges in 0 have a
costof flu I I
I
y
senti
if G has a H Ccat ofopt.to or I v I s f w 1
if c has a tour of arts f Iv l
G has aH
f
ti is lengthofjob I
I
I I k
r
t
a EI
E
This is a z approximation
times
H H A 11 Hat
m resources
7 7 2tm
to Etna
ti f k T’t
Ti c
tj flat
Thi is a e 5
approximations
ateachstep we are placing 2 nodes
in the set Theopt so1
needs atleast oneof them
Q doo Y
ftp.n.o
o
0
i
Yes
RHSvector
coefficient
matrix unknowns
he
E t
rain
t
Cajon
O
O
O
ai is a decisionvariablefor
each no de e e V
ai 0 if s
ki I i E S
e f Ini t ai s l
ai ai
Minimize Swine
Subject to
Kitaj 21 force p
EE
mm
ooyrfoos
0
0
hkpfw
WC53_
ai’t o i f s
ri’t p i ES
r
Say Ss se f i e V i ni la
in
Say S is our approx So l
W S f 2 Ufp
we s w Cs
W S f 2 W S
This is a z approximation
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Before Lecture Notes – 14
Before Lecture Notes – 15