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ca r ni z i X
a n

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Satisfiability
problem

SAT

General Form

3 SAT

0 0
Plan Given an instance of 3 SAT

w K clauses Build agraph
That has an indep setof Gi kiff the 3 SAT instance is satisfiable

N

x van a

a
d

C R V R2 Vaz
u I ATV away

C3 Tea V kEUR3

Gil I

Do

a ai

set a e I

set ai O

set ai toeither Eor I

r

E

if X c NP and for all Ye NP
YEp X then X is the hardestproblem

NP

3 SAT has been proven to be
such a problem

such aproblem is calledNP complete

Transitivity

42SPY and YEpX
Him 2 Epix

3s ATSpindpsut spvertexcoverspseta.ru

P NP Don’t know

Basic strategy to show
a problem Is N D Complete

f Prove X E N P

2 Choose a problem Y that is
known to be ND Complete

2 Prove that I Ep E

NPhard Setof problems
that are at least as hard as

ND Complete problems

g

NPcomplete

Inly
a
handfulof

problems

Traveling Salesman Problem TSP

Hamiltonian Cycle

Distort

0 0

Def A cycle C in G is a
Hamiltonian Cycle if itvisitseachvertex exactly once

Problem statement

Given an undirectedgraph G is there
a Hamiltonian cycle in G

k
0

Show that the HamiltonianCycle
Problem is NP complete

1 Show Ham Cycle is in N P
certificate ordered list ofnodes

out he a
certifier e All nodes appearing

check there is an edge
between needs adjacentofnodes

edgebetween last
firstnodes

0 no repeating nodes
is the bit

2 Choose a problem that
we
already know is NP complete

choose

vertexcoversshow that vertexconaspHam cycle

Ew.DK Eu v.D
YU 2 o o

n
a

1

Tu if.f
G

0 I

YU 6 v.v
I l
f

I
I

Fu TO 8 8

O

Beginning
Fpath

i

i

day
path

Hoz
w

tE

G

O

X Y X W X V X z A
G

in

G k 2

no

Proof A suppose that G Cv E has
a vertex coverofSgi E let the
vertex cover setbe

S Ui Uz Uh

we will identify neighborsofUi as
shown here

f
i

i

gguit

Form a Ham Cycle in G byfollowing
the nodes in G in this order i
start at s and go to
u vi if u vi 6J
u iUf if u U G
e

v That if u Thi g

O

Then go to Sz and follow the nodes
Uz Uz I Uz Uz 6

Uz UE I Uz UE G

a yds if Luz vis o
Then go to Sz

I

uh vii I un Uh 63
Uh UE I Uh uh 6

Ye u if fun under D
Then return back to s

B Suppose G has a Hamiltonian
cycle C then the set

S uj ell i Csg Cui ui D e C

for some k j Ek

will be a vortex cover set in G

O 0O_

Before Lecture Notes – 13
Before Lecture Notes – 14