meljun cortes -- automata theory lecture - 23

7/31/2019 MELJUN CORTES -- AUTOMATA THEORY LECTURE - 23

1/33

CSC 3130: Automata theory and formal languages

Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130

NP-complete problems

Fall 2008


2/33

Polynomial-time reductions

Language Lpolynomial-time reduces to Lif

there exists a polynomial-time computable

map R that takes an instancexofL intoinstanceyofLs.t.

x L if and only if y L

L L(IS) (CLIQUE)

x y(G, k) (G, k)

x L y L(G has IS of size k) (Ghas clique of size k)

R


3/33

The Cook-Levin Theorem

Every L NP reduces to SAT

SAT= {f: fis a satisfiable Boolean formula}

(x1x2 ) (x2x3x4) (x1)f=

is satisfiable

x1 = Tx2 = Fx3 = Tx4 = T

(x1x2 ) (x1) (x2)f=

is not satisfiable


4/33

NP-hardness

Language L is NP-hard if every Lin NPreduces to L

Intuitively, NP-hard means harder than allof NP

The Cook-Levin Theorem says

SAT is NP-hard

P

SAT

NP


5/33

NP-complete

L is NP-complete ifL is NP-hard and L NP

Intuitively, NP-complete means hardest inNP

Recall that SAT NP, so SAT is NP-complete

If SAT P, then P = NP

P

SAT

NP


6/33

Proof of Cook-Levin Theorem

To prove it, we have to describe areduction R:

Every L NP reduces to SAT

w Boolean formula f

w L fis satisfiable

R


7/33


All we know about L: It has a poly-time NTMM

Lets look at computation tableau ofM on inputw

M w w1 w2q0

qacc

S-th configurationsymbol at time T

S

T

Since M is nondeterministic,there may be many possibletableaus


8/33


w1 w2q0

qacc

S

T

n = length of input w

height of tableau is p(n) for some polynomialp

width is at most p(n)

kpossible tableau symbols

xT, S, u =true,if the (T, S) cell of

tableau contains u

if notfalse,

u

1 S p(n)

1 Tp(n)

1 u k


9/33


We will design a formula fsuch that:

w Boolean formula f

w L fis satisfiable

R

variables off:

assignment toxT, S, u

way to fill up the tableau

satisfying assignment accepting computation tableau

fis satisfiable

xT, S, u

M accepts w


10/33


We want to construct (in time poly(n)) aformula f:

xT, S, u =true,if the (T, S) cell of

tableau contains u

if notfalse,

1 S p(n)1 Tp(n)

1 u k

f(x1, 1, 1, ...,xp(n),p(n), k) =

true,if thexs represent a validaccepting tableau

if notfalse,


11/33


w1 w2q0

qacc

S

T

u

f= fcell f0 fmove facc

fcell:Every cell contains

exactly one symbol

The first row isq0w1w2...wk...

f0:

fmove:The moves between rowsfollow the transitions ofM

facc: qacc appears somewhere in the last

row


12/33


Desired meaning

Implementation:

fcell: Every cell contains exactly onesymbol

fcell = fcell1, 1 ... fcellp(n),p(n)where

fcellT, S = (xT, S, 1 ... xT, S, k)

(xT, S, 1xT, S, 2)

(xT, S, 1xT, S, 3)

...( xT, S, k-1xT, S, k)

at least one symbol

no two symbols

or: Exactly one ofxS, T, 1 ... xS, T, kis true


13/33


Desired meaning

Implementation:

f0:The first row is q0w1w2...wk...

facc: qacc appears somewhere in the last

row

f0 =x1, 1, q0x1, 1, w1x1, 1, w2 ... x1,p(n),

facc =xp(n), 1, qaccxp(n), 2, qacc ... xp(n),p(n), qacc


14/33

Valid and invalid windows

6c3a0t0

0c6a0p0

0

6t3t0u0

0t6t0u0

0

valid window

invalid window

6t3t0u0

0t6t0q3

0

valid window

6t3q3u0

0t6a0q7

0

valid if(q3, u) = (q7, a, R)

6q3t0u0

0k6t0q0

0

invalid window

6c3a0t0

0b6a0t0

0

valid window


15/33


Desired meaning

Implementation:

fmove:The moves betweenrowsfollow transitions of

M

a aq0

qacc

fmove = fmove1, 1 ... fmovep(n)-3,p(n)-3

b

q3b a b

b q7c b

fmove2, 2

fmoveT, S =over all

(xT, S, a1 xT, S+1, a2 xT, S+2, a3

xT+1, S, a4 xT+1, S+1, a5 xT+2, S+1,

a6)

valid windows

a1a2a3a4a5a6


16/33

Other NP-complete problems

CLIQUE = {(G, k): G is a graph with a clique ofkvertices}IS = {(G, k): G is a graph with an independent set ofkvertices}

VC = {(G, k): G is a graph with a vertex cover ofkvertices}

CLIQUE, IS and VC are NP-complete

SAT

NP

CLIQUE IS VC


17/33

Proving NP-hardness

To show L is NP-hard, it is enough toreduce from some Lwe already know isNP-hard

For now we can take L= SAT

To show L is NP-complete,we also need to argue that

L is in NPThis is usually the easy part

roadmap:

SAT

3SAT

IS

CLIQUE VC


18/33

3SAT

SAT= {f: fis a satisfiable Boolean formula}3SAT= {f: fis a satisfiable Boolean formula in

conjunctive normal form withat most 3 distinct literals per clause}

CNF: AND of ORs of literals

literal:xi orxi

(conjunctive normal form)

3CNF: CNF with 3 lit/clause

(x1x2 ) (x2x3x4) (x1)clauseliterals

(x2(x1x2 ))(x1(x1x2 ))gates


19/33

NP-hardness of 3SAT

Theorem

Proof: We describe a reduction R from SAT

Boolean formula f 3CNF formula f

fis satisfiable

R

fis satisfiable

3SAT is NP-hard


20/33

Reducing SAT to 3SAT

Example: f=(x2(x1x2 ))(x1(x1x2 ))

e give extra variables to every gate (wire)

x3

x6 x7

x8 x9

x4 x5

x10AND

OR NOT

AND

NOT OR

AND

NOT

x2 x1 x2 x1 x1x2

x7 =x4x5x4x5x7T T T T

T T F F

T F T F

T F F T

F T T F

F T F T

F F T F

F F F T

(x4x5x7)(x4x5x7)

(x4x5x7)

(x4x5x7)

(x4x5x7)(x4x5x7)(x4x5x7)(x4x5x7)


21/33


Step 1: Add variablexn+j for each gate Gj inf

Step 2: Write 3CNF fj for each gate Gj,j ={1, ..., t}

Step 3: Output

Boolean formula f 3CNF formula fR

f= fn+1 fn+2 ... ftxn+t

requires thatoutput offis true


22/33


Every satisfying assignment offextendsuniquely to a satisfying assignment off

Conversely, every satisfying assignment offmust contain a satisfying assignment off

Boolean formula f 3CNF formula fR

fis satisfiablefis satisfiable


23/33

Independent set

Theorem

IS = {(G, k): G is a graph with an independentset ofkvertices}

1 2

3 4

An independent setis a subset ofvertices so that nopair is connected

{1, 2}, {1, 3}, {4} areindependent sets

IS is NP-hard

SAT

3SAT

IS

CLIQUE VC


24/33

Reducing 3SAT to IS

Proof: We describe a reduction from 3SATto IS

3SAT= {f: fis a satisfiable Boolean formula in 3CNF}

IS = {(G, k): G is a graph with an independent set ofkvertices}

3CNF formula f (G, k)

G has an independentset of size k

R

fis satisfiable


25/33

Reducing 3SAT to IS

Example: f=(x1x2 ) (x2x3x4) (x1)

TT

TF

FT

FF

TTT

TTF

TFT

TFF

FTT

FTF

FFTFFF

T

F

Put an edge for every inconsistency

all interconnected

x2x3x4x1x2

x1

etc.


26/33

Reducing 3SAT to IS

Example: f=(x1x2 ) (x2x3x4) (x1)

TT

TF

FF

TTT

TTF

TFT

TFF

FTT

FFTFFF

T

x2x3x4x1x2

x1

G

satisfying assignment off

IS of size 3 in G

x1 = Tx2 = Fx3 = Tx4 = T

edges = inconsistencies

any IS of size 3 in G

satisfying assignment off


27/33

Reducing 3SAT to IS

G has a vertex for every clause Ci offandevery satisfying assignment ofCi

G has an edge between any two verticesthat represent inconsistent assignments

kis the number of clauses in f

3CNF formula f (G, k)R


28/33

Reducing 3SAT to IS

Every satisfying assignment offgives anindependent set of size kin G

Conversely, from every IS of size kin G wecan extract a consistent and satisfyingassignment off

3CNF formula f (G, k)

G has an IS of size k

R

fis satisfiable


29/33

Vertex cover

SAT

3SAT

IS

CLIQUE VC

Theorem

VC is NP-hard

A vertex cover is aset of vertices thattouches (covers) alledges

{2, 4}, {3, 4}, {1, 2, 3}are vertex covers

1 2

3 4

VC = {(G, k): G is a graph with a vertex cover of size k}


30/33

Reducing IS to VC

Proof: We describe a reduction from IS toVC

Example

(G, k)

Ghas a VC of size k

R(G, k)

G has an IS of size k

1 2

3 4

, {1}, {2}, {3},{4},{1, 3}, {2, 3}

{2, 4}, {3, 4},{1, 2, 3}, {1, 2,4},{1, 3, 4}, {2, 3,4},

{1, 2, 3, 4}

independent setsvertex covers


31/33

Reducing IS to VC

Claim

Proof

S is an independent set ofG if andonly ifS is a vertex cover ofG

S is an independent set ofG

no edge has both endpoints in S

every edge has an endpoint in S

S is a vertex cover ofG


32/33

Reducing IS to VC

Set G = G, k= n k(n = number of

vertices)

by previous Claim.

(G, k)R(G, k)

Ghas a VC of size kG has an IS of size k

The ubiquity of NP complete


33/33

The ubiquity of NP-completeproblems

We saw a few examples of NP-completeproblems, but there are many more

A surprising fact of life is that most CSproblems are either in P or NP-complete

A 1979 book by Garey and Johnson

lists 100+ NP-complete problems

meljun cortes -- automata theory lecture - 23

Documents