Some Optimal Inapproximability Results

David Arnon March 3, 2005 Inapproximability Seminar – 2005

Some Optimal Inapproximability Results

Johan Håstad

Royal Institute of Technology, Sweden

2002

Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005

Bound Summary Problem Upper Lower

E3-LIN-2 2 2 –

E3-SAT 8/7 8/7 –

E3-LIN-p p p –

E3-LIN- || ||

E4-Set Splitting 8/7 8/7 –

E2-LIN-2 1.1383 12/11 –

E2-SAT 1.0741 22/21 –

Max-Cut 1.1383 17/16 –

Max-di-Cut 1.164 12/11 –

Vertex Cover 2 7/6 –


Overview

gap(,1)

LABELCOVER

gap(½+, 1)

E3-LIN-2

gap(⅞+, 1)

3SAT

Long Code + Håstad’s LABELCOVER Junta testing

3SAT gap(c,1)

3SATPCP theorem

ParallelRepetitionTheorem

4-gadget


Hardness of MAX-E3-SAT

gap(½+, 1)-E3-LIN-2 can be reduced togap(⅞+¼ , 1¼)-E3-SAT.



xyz = 1

xyz = 1

(xVyVz),(xVyVz),(xVyVz),(xVyVz)

(xVyVz),(xVyVz),(xVyVz),(xVyVz)

gap(½+, 1)-E3-LIN-2 can be reduced togap(⅞+¼ , 1¼)-E3-SAT.

4-gadget


Overview

gap(,1)

LABELCOVER

gap(½+, 1)

E3-LIN-2

gap(⅞+, 1)

3SAT


3SAT gap(c,1)

3SATPCP theorem


4-gadget


LABEL COVER

An instance of the LABEL COVER problem is denoted by: L(G(V,W,E) ,[n] ,[m] ,) where:

G(V,W,E) is a regular bipartite graph. [n], [m] are sets of labels for V, W.

{wv}(v,w)E

For every edge (v,w) wv is a map wv:[m][n]


LABEL COVER

A labeling V[n], W[m]satisfies wv if wv( (w)) = (v).

For an instance L, The maximum fraction of constraints wv that can be satisfied by any labeling is denoted by OPT(L).

The goal: Find a labeling that satisfies OPT(L) of the constraints.


PCP Theorem c(0,1) s.t.

gap(c,1)-MAX-E3-SAT is NP-hard.

For that c:The gap-LABEL COVER problem:gap(⅓(2+c),1)-L(G(V,W,E) ,[2] ,[7] ,) is NP-hard.


V Vk

[n] [n]k

Given L(G(V,W,E) ,[n] ,[m] ,)define Lk(G(V,W,E) ,[n] ,[m] ,) :

V Vk W Wk

[n] [n]k [m] [m]k

(v,w)E for v=(v1,…,vk) w=(w1,…,wk) iff i[k] (vi,wi)E

For every wv define:wv(m1,…,mk) = (w1v1

(m1),…,wkvk(mk))

LABEL COVER - Repetition


Raz’s Parallel Repetition Theorm

Given a LABEL COVER problem L,if OPT(L) = c < 1 then there exists cc < 1that depends only on c, n & m s.t.OPT(Lk) cc

k .


LABEL COVER - Conclusion

For every > 0 there are N, M s.t.the gap-LABEL COVER problem:gap(,1)-L(G(V,W,E) ,[N] ,[M] ,)is NP-hard


Overview

gap(,1)

LABELCOVER

gap(½+, 1)

E3-LIN-2

gap(⅞+, 1)

3SAT


3SAT gap(c,1)

3SATPCP theorem


4-gadget


The Long Code

For every i[n] the Long CodeLCi :{1,1}[n] {1,1} is defined.For every f:[n] {1} :LCi (f ) f(i)

LCi X{i}


Fourier Analysis - Reminder

Linear functions: [n] X(x)ixi

Inner Product Space: <A,B> Ex[A(x)B(x)]

<X,X> =

{X}[n] is an orthonormal basis for {[n]RR



Every A:{[n] { can be written as: A = [n]ÂX

Â[n] are called the Fourier coefficients of A.

Parseval’s identity:for any boolean function A we have [n]Â

= 1



Â= <A,X>

Prx[A(x) = X(x)] = ½ + ½Â

Â= Ex[A(x)]

X{i}(x) = xi = LCi(x) (Dictatorship)


Testing the Long Code Linearity Test

Choose f,g{[n] at random.

Check if: A(f)A(g) = A(fg)

Perfect completeness.


Testing the Long Code Junta Test, parameterized by Choose f,g{[n] at random. Choose {[n] by setting:

x[n] x

Check if: A(f)A(g) = A(fg)

1 with probability

1 with probability 1


Standard Written Assignmentfor the LABEL COVER

Given a LABEL COVER problem L(G(V,W,E) ,[n] ,[m] ,)And an assignment that satisfy all the constraints,

The SWA() contains for every vV theLong Code of it’s assignment LC(v)

and for every wW it’s LC(w).


Testing the SWA – L2() Håstad’s LABEL COVER Test

Given: LABEL COVER problem L(G(V,W,E) ,[n] ,[m] ,) A supposed SWA for it.

Choose (v,w)E at random. Denote (the supposed) LC(v) by A

and (the supposed) LC(w) by B.



Choose f{[n] at random. Choose g{[m] at random. Choose {[m] by setting:

x[m] x

Check if: A(f)B(g) = B((fwvg )

1 with probability

1 with probability 1



Completeness: 1



Completeness: 1

Soundness: For any LABEL COVER problem L

and any > 0, if the probability thattest L2() accepts is ½(1+ ) thenthere is a assignment that satisfy 4

of L`s constraints.


Hardness of MAX-E3-LIN2

For any 0gap(½+, 1)-E3-LIN-2 is NP-hard.


Testing the SWA - Folding

In order to ensure that A is balanced we forceA(f ) = A(f ) by reading only half of A:

A(f ) = A(f ) if f(1) = 1

A(f ) if f(1) =



Ew,v[ÂB12

Ew,v[ÂB

^

^


x-½ e-x/2


x-½ e-x/2

e-x 1-x


Hardness of MAX-E3-LIN2

For any 0 it is NP-hard to approximateMAX-E3-LIN-2 within a factor of 2.

MAX-E3-LIN-2 is non-approximable beyond the random assignment threshold.


Overview

gap(,1)

LABELCOVER

gap(½+, 1)

E3-LIN-2

gap(⅞+, 1)

3SAT


3SAT gap(c,1)

3SATPCP theorem


4-gadget



For any 0 it is NP-hard to approximateMAX-E3-SAT within a factor of 8/7.


FIN

Some Optimal Inapproximability Results

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