Inapproximability Seminar – 2005 David Arnon March 3, 2005 Some Optimal Inapproximability Results...
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Transcript of Inapproximability Seminar – 2005 David Arnon March 3, 2005 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 –
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Hardness of MAX-E3-SAT
gap(½+, 1)-E3-LIN-2 can be reduced togap(⅞+¼ , 1¼)-E3-SAT.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Hardness of MAX-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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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]
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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 .
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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}
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Fourier Analysis - Reminder
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Fourier Analysis - Reminder
Â= <A,X>
Prx[A(x) = X(x)] = ½ + ½Â
Â= Ex[A(x)]
X{i}(x) = xi = LCi(x) (Dictatorship)
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Testing the Long Code Linearity Test
Choose f,g{[n] at random.
Check if: A(f)A(g) = A(fg)
Perfect completeness.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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).
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Testing the SWA – L2() Håstad’s LABEL COVER Test
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Testing the SWA – L2() Håstad’s LABEL COVER Test
Completeness: 1
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Testing the SWA – L2() Håstad’s LABEL COVER Test
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.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Hardness of MAX-E3-LIN2
For any 0gap(½+, 1)-E3-LIN-2 is NP-hard.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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) =
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Testing the SWA – L2() Håstad’s LABEL COVER Test
Ew,v[ÂB12
Ew,v[ÂB
^
^
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
x-½ e-x/2
e-x 1-x
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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.
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
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
Some Optimal Inapproximability Results – Johan Håstad Inapproximability Seminar – 2005
Hardness of MAX-E3-SAT
For any 0 it is NP-hard to approximateMAX-E3-SAT within a factor of 8/7.