NP-complete Languages

Costas Busch - LSU 1

NP-complete Languages


Polynomial Time Reductions

Polynomial Computable function : f

such that for any string computesin polynomial time:

)(wfwThere is a deterministic Turing machine

)|(| kwO

M


)|(||)(| kwOwf

since, cannot use more than tape spacein time

M)|(| kwO

)|(| kwO

Observation:

the length of is bounded)(wf


Language is polynomial time reducible to language

if there is a polynomial computable function such that:

f

BwfAw )(

A

B

Definition:


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


Example of a polynomial-time reduction:

We will reduce the

3CNF-satisfiability problemto the

CLIQUE problem


3CNF formula:

)()()()( 654463653321 xxxxxxxxxxxx

Each clause has three literals

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

w wLanguage:

literal

clause

variable or its complement


A 5-clique in graph

CLIQUE = { : graph contains a -clique}

kG, Gk

G

Language:


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

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

Transform formula to graph


)()()()( 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:


)()()()( 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


)()()()( 432321421421 xxxxxxxxxxxx

1x

2x

1x 2x 4x

1x

2x

3x

2x 4x

4x

3x

Resulting Graph


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


We define the class of NP-completelanguages

Decidable

NP-complete

NP-complete Languages

NP


A language is NP-complete if:

• is in NP, and

• Every language in NP is reduced to in polynomial time

L

L

L


Observation:

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

NPP


NP-complete

NP

P ?

Decidable


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


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:


0q

iq jq

… … …

… accept

accept

reject

reject

kn

(deepest leaf)

depth

All computations of on stringw

nw ||

M


0q

iq jq

… … …

… accept

accept

reject

reject

kn

(deepest leaf)

depth

Consideran acceptingcomputation


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


w

n

knkn

kn2Maximum working space area on tapeduring time steps kn

Machine TapeM


##

#

##

#

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

……


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

t1r1

C

Tableau Alphabet

Finite size (constant))1(|| OC


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


##

##

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


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


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


tjisjitsCtssjiCsji

xxx ,,,,,,,, allcell

Size of :cell

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

)( 2knO


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


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


accept makes sure that the computationleads to acceptance

qjijix ,,

F q all, allaccept

An accept state should appear somewherein the tableau

Accepting states


Size of :accept

qjijix ,,

F q all, allaccept

)()32( 2kkk nOnn


move makes sure that the tableaugives a valid sequence of configurations

move is expressed in terms of legal windows


Tableau

a 1q b2q a c

Window

2x6 area of cells


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


Possible illegal windows

Rba ,

Lcb ,1q

Rab ,

2q b

2q

a1q

ba

b

b

b

1q

1q

2q

aaa

aaa


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


ija 1q b2q a c

(is legal)

cjiajiqji

bjiqjiaji

xxxxxx

,2,1,1,1,,1

,,,1,,,

2

1

Formula:


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



Size of :),( wM

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

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




we have that:


is constructed in polynomial time


),( wM

Since,

and

Lis polynomial-time reducible to SAT

END OF PROOF


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

),( wM


Already CNFNOT CNF

But can be converted to CNFusing distributive laws


Distributive Laws:

)()()( RPQPRQP

)()()( RPQPRQP


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

),( wM

laaa 21

)()()( 13342231121 lll zazzazzazzaa

convert


From Observations 1 and 2:

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

(they are known NP languages)


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


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)

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NP-complete Languages