np-complete languages

Costas Busch - LSU 1

NP-complete Languages

Costas Busch - LSU 2

Polynomial Time Reductions

Polynomial Computable function : f

such that for any string computesin polynomial time:

)(wfwThere is a deterministic Turing machine

)|(| kwO

M

Costas Busch - LSU 3

)|(||)(| kwOwf

since, cannot use more than tape spacein time

M)|(| kwO

)|(| kwO

Observation:

the length of is bounded)(wf

Costas Busch - LSU 4

Language is polynomial time reducible to language

if there is a polynomial computable function such that:

f

BwfAw )(

A

B

Definition:

Costas Busch - LSU 5

Suppose that is polynomial reducible to .If then .

Theorem:

PBA B

PAProof:

Machine to decide in polynomial time:AOn input string :w 1. Compute )(wf

Let be the machine that decides BM

2. Run on input )(wfM

in polynomial time

3. If acccept Bwf )( w

M

Costas Busch - LSU 6

Example of a polynomial-time reduction:

We will reduce the

3CNF-satisfiability problemto the

CLIQUE problem

Costas Busch - LSU 7

3CNF formula:

)()()()( 654463653321 xxxxxxxxxxxx

Each clause has three literals

3CNF-SAT ={ : is a satisfiable 3CNF formula}

w wLanguage:

literal

clause

variable or its complement

Costas Busch - LSU 8

A 5-clique in graph

CLIQUE = { : graph contains a -clique}

kG, Gk

G

Language:

Costas Busch - LSU 9

Theorem: 3CNF-SAT is polynomial time reducible to CLIQUE

Proof: give a polynomial time reductionof one problem to the other

Transform formula to graph

Costas Busch - LSU 10

)()()()( 432321421421 xxxxxxxxxxxx

Clause 2

Clause 1

Clause 3

1x

2x

1x 2x 4x

1x

2x

3x

2x 4x

4x

3x

Transform formula to graph. Example:

Clause 4

Create Nodes:

Costas Busch - LSU 11

)()()()( 432321421421 xxxxxxxxxxxx

1x

2x

1x 2x 4x

1x

2x

2x 4x

4x

3x

3x

Add link from a literal to a literal in everyother clause, except the complement

Costas Busch - LSU 12

)()()()( 432321421421 xxxxxxxxxxxx

1x

2x

1x 2x 4x

1x

2x

3x

2x 4x

4x

3x

Resulting Graph

Costas Busch - LSU 13

1001

4

3

2

1

xxxx

1)()()()( 432321421421 xxxxxxxxxxxx

1x

2x

1x 2x 4x

1x

2x

3x

2x 4x

4x

3x

The formula is satisfied if and only ifthe Graph has a 4-clique End of Proof

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We define the class of NP-completelanguages

Decidable

NP-complete

NP-complete Languages

NP

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A language is NP-complete if:

• is in NP, and

• Every language in NP is reduced to in polynomial time

L

L

L

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Observation:

If a NP-complete languageis proven to be in P then:

NPP

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NP-complete

NP

P ?

Decidable

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Cook-Levin Theorem: Language SAT (satisfiability problem) is NP-complete

Proof:

Part2: reduce all NP languages to the SAT problem in polynomial time

Part1: SAT is in NP(we have proven this in previous class)

An NP-complete Language

Costas Busch - LSU 19

For any string we will construct in polynomial time a Boolean expression ),( wM

esatisfiabl is ),( wMLw

Take an arbitrary language NPL

We will give a polynomial reduction of to SAT L

Let be the NonDeterministicTuring Machine that decides in polyn. time

ML

w

such that:

Costas Busch - LSU 20

0q

iq jq

… … …

… accept

accept

reject

reject

kn

(deepest leaf)

depth

All computations of on stringw

nw ||

M

Costas Busch - LSU 21

0q

iq jq

… … …

… accept

accept

reject

reject

kn

(deepest leaf)

depth

Consideran acceptingcomputation

Costas Busch - LSU 22

0q

iq

…

accept

jq

mq

aqknlal

ni

n

q

qq

11

21

210

initial state

accept state

Computation path Sequence of Configurations

nw 21

:xnk

:1:2

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w

n

knkn

kn2Maximum working space area on tapeduring time steps kn

Machine TapeM

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##

#

##

#

0q 1iq

2 n1 2 n

:1:2

# #:x aq

:kn

32 kn

Tableau of Configurations

Accept configuration

indentical rows1 2 3 kn

1l

1 2 3 1

l knaq

kn kn

……

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}q,,{q},,{}#{states} of {setalphabet} {tape}#{

t1r1

C

Tableau Alphabet

Finite size (constant))1(|| OC

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For every cell position ji ,andfor every symbol in tableau alphabet Cs

Define variable sjix ,,

Such that if cell contains symbolThen Else

ji , s1,, sjix0,, sjix

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##

##

0q 1iq

2 n1 2 n

:1:2

Examples:

1#,1,1 x

3kn

1,3,2 i

k qnx

0,1,1 x 0#,3,2 kn

x

Costas Busch - LSU 28

moveacceptstartcell),( wM

),( wM is built from variables sjix ,,

When the formula is satisfied,it describes an accepting computation in the tableau of machine on inputM w

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cell makes sure that everycell in the tableau contains exactly one symbol

tjisjitsCtssjiCsji

xxx ,,,,,,,, allcell

Every cell containsat least one symbol

Every cell containsat most one symbol

Costas Busch - LSU 30

tjisjitsCtssjiCsji

xxx ,,,,,,,, allcell

Size of :cell

||C 2||C( )kk nkn )2(

)( 2knO

Costas Busch - LSU 31

start makes sure that the tableaustarts with the initial configuration

#,22,1,22,1,3,1

,2,1,3,1,2,1

,1,1,2,1#,1,1start

10

kkk

nkkk

k

nnnn

nnnqn

n

xxx

xxxxxx

Describes the initial configurationin row 1 of tableau

Costas Busch - LSU 32

Size of :start

#,32,1,22,1

,3,1,2,1,1,1

,2,1#,1,1start

10

kk

kkk

nn

nqnn

xx

xxxxx

)(32 kk nOn

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accept makes sure that the computationleads to acceptance

qjijix ,,

F q all, allaccept

An accept state should appear somewherein the tableau

Accepting states

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Size of :accept

qjijix ,,

F q all, allaccept

)()32( 2kkk nOnn

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move makes sure that the tableaugives a valid sequence of configurations

move is expressed in terms of legal windows

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Tableau

a 1q b2q a c

Window

2x6 area of cells

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Possible Legal windows

Rba ,

Lcb ,1q

Rab ,

2qa

a 1q b2q a

aa

c

bb

b

b

1q

1q

2q

2qaaa

aa a

a

Legal windows obey the transitions

Costas Busch - LSU 38

Possible illegal windows

Rba ,

Lcb ,1q

Rab ,

2q b

2q

a1q

ba

b

b

b

1q

1q

2q

aaa

aaa

Costas Busch - LSU 39

legal) is j)(i, window(ji, allmove

ij

aa 1q b2q a

aac b

b 1q1q2qaa

aa

j

i ij

window (i,j) is legal:

((is legal) (is legal) (is legal))

all possible legal windowsin position ),( ji

Costas Busch - LSU 40

ija 1q b2q a c

(is legal)

cjiajiqji

bjiqjiaji

xxxxxx

,2,1,1,1,,1

,,,1,,,

2

1

Formula:

Costas Busch - LSU 41

Size of : move

Number of possible legal windows in a cell i,j: 6||Cat most

Size of formula for a legal window in a cell i,j: 6

Number of possible cells: kk nn )32(

x

x )()32(|| 26 kkk nOnnC

Costas Busch - LSU 42

moveacceptstartcell),( wM

Size of :),( wM

)( 2knO )( knO )( 2knO )( 2knO )( 2knO

polynomial in nit can also be constructed in time )( 2knO

Costas Busch - LSU 43

esatisfiabl is ),( wMLw

moveacceptstartcell),( wM

we have that:

Costas Busch - LSU 44

is constructed in polynomial time

esatisfiabl is ),( wMLw

),( wM

Since,

and

Lis polynomial-time reducible to SAT

END OF PROOF

Costas Busch - LSU 45

Observation 1:The formula can be convertedto CNF (conjunctive normal form) formulain polynomial time

),( wM

moveacceptstartcell),( wM

Already CNFNOT CNF

But can be converted to CNFusing distributive laws

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Distributive Laws:

)()()( RPQPRQP

)()()( RPQPRQP

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Observation 2:The formula can alsobe converted to a 3CNF formulain polynomial time

),( wM

laaa 21

)()()( 13342231121 lll zazzazzazzaa

convert

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From Observations 1 and 2:

CNF-SAT and 3CNF-SAT are NP-complete languages

(they are known NP languages)

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Theorem:If: a. Language is NP-complete b. Language is in NP c. is polynomial time reducible to

A

A BB

Proof:Any language in NPis polynomial time reducible to .Thus, is polynomial time reducible to

AB

(sum of two polynomial reductions,gives a polynomial reduction)

L

L

Then: is NP-completeB

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Corollary: CLIQUE is NP-complete

Proof:

b. CLIQUE is in NP (shown in last class)c. 3CNF-SAT is polynomial reducible to CLIQUE

a. 3CNF-SAT is NP-complete

Apply previous theorem withA=3CNF-SAT and B=CLIQUE

(shown earlier)