Download presentation

Presentation is loading. Please wait.

Published byJanessa Gentile Modified over 2 years ago

1
Inapproximability Seminar – 2005 David Arnon March 3, 2005 Some Optimal Inapproximability Results Johan Håstad Royal Institute of Technology, Sweden 2002

2
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Bound Summary ProblemUpperLower E3-LIN-22 2 – E3-SAT8/7 8/7 – E3-LIN-pp p – E3-LIN- ||||| | E4-Set Splitting8/7 8/7 – E2-LIN-21.1383 12/11 – E2-SAT1.0741 22/21 – Max-Cut1.1383 17/16 – Max-di-Cut1.164 12/11 – Vertex Cover2 7/6 –

3
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Overview gap( ,1) L ABEL C OVER gap(½+ , 1 ) E3-LIN-2 gap(⅞+ , 1 ) 3SAT Long Code + Håstad’s L ABEL C OVER Junta testing 3SAT gap(c,1) 3SAT PCP theorem Parallel Repetition Theorem 4-gadget

4
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Hardness of M AX -E3-S AT gap(½+ , 1 )-E3-LIN-2 can be reduced to gap(⅞+¼ , 1 ¼ )-E3-SAT.

5
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Hardness of M AX -E3-S AT xyz = 1 xyz = 1 (x V y V z),(x V y V z),(x V y V z),(x V y V z) gap(½+ , 1 )-E3-LIN-2 can be reduced to gap(⅞+¼ , 1 ¼ )-E3-SAT. 4-gadget

6
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Overview gap( ,1) L ABEL C OVER gap(½+ , 1 ) E3-LIN-2 gap(⅞+ , 1 ) 3SAT Long Code + Håstad’s L ABEL C OVER Junta testing 3SAT gap(c,1) 3SAT PCP theorem Parallel Repetition Theorem 4-gadget

7
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad L ABEL C OVER An instance of the L ABEL C OVER 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]

8
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad L ABEL C OVER 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.

9
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad PCP Theorem c (0,1) s.t. gap(c,1)-M AX -E3-S AT is NP-hard. For that c: The gap-L ABEL C OVER problem: gap(⅓(2+c),1)- L (G(V,W,E),[2],[7], ) is NP-hard.

10
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad V V k [n] [n] k Given L (G(V,W,E),[n],[m], ) define L k (G(V,W,E),[n],[m], ) : V V k W W k [n] [n] k [m] [m] k (v,w) E for v=(v 1,…,v k ) w=(w 1,…,w k ) iff i [k] (v i,w i ) E For every wv define: wv (m 1,…,m k ) = ( w 1 v 1 (m 1 ),…, w k v k (m k )) L ABEL C OVER - Repetition

11
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Raz’s Parallel Repetition Theorm Given a L ABEL C OVER problem L, if OPT( L ) = c < 1 then there exists c c < 1 that depends only on c, n & m s.t. OPT( L k ) c c k.

12
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad L ABEL C OVER - Conclusion For every > 0 there are N , M s.t. the gap-L ABEL C OVER problem: gap( , 1 )- L (G(V,W,E),[N ],[M ], ) is NP-hard

13
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Overview gap( ,1) L ABEL C OVER gap(½+ , 1 ) E3-LIN-2 gap(⅞+ , 1 ) 3SAT Long Code + Håstad’s L ABEL C OVER Junta testing 3SAT gap(c,1) 3SAT PCP theorem Parallel Repetition Theorem 4-gadget

14
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad The Long Code For every i [n] the Long Code LC i :{ 1,1} [n] { 1,1} is defined. For every f:[n] { 1} : LC i (f ) f(i) LC i X {i}

15
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Fourier Analysis - Reminder Linear functions: [n] X (x) i x i Inner Product Space: E x [A(x)B(x)] = R { X } [n] is an orthonormal basis for { [n] R

16
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Fourier Analysis - Reminder Every A:{ [n] { can be written as: A = [n] Â X Â [n] are called the Fourier coefficients of A. Parseval’s identity: for any boolean function A we have [n] Â = 1

17
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Fourier Analysis - Reminder Â = Pr x [A(x) = X (x)] = ½ + ½Â Â = E x [A(x)] X {i} (x) = x i = LC i (x)(Dictatorship)

18
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the Long Code Linearity Test Choose f,g { [n] at random. Check if: A(f)A(g) = A(fg) Perfect completeness.

19
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad 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

20
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Standard Written Assignment for the L ABEL C OVER Given a L ABEL C OVER problem L (G(V,W,E),[n],[m], ) And an assignment that satisfy all the constraints, The SWA( ) contains for every v V the Long Code of it’s assignment LC (v) and for every w W it’s LC (w).

21
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA – L 2 ( ) Håstad ’ s L ABEL C OVER Test Given: L ABEL C OVER 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.

22
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA – L 2 ( ) Håstad ’ s L ABEL C OVER 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((f wv g ) 1 with probability 1 with probability 1

23
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA – L 2 ( ) Håstad ’ s L ABEL C OVER Test Completeness: 1

24
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA – L 2 ( ) Håstad ’ s L ABEL C OVER Test Completeness: 1 Soundness: For any L ABEL C OVER problem L and any > 0, if the probability that test L 2 ( ) accepts is ½(1+ ) then there is a assignment that satisfy 4 of L `s constraints.

25
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Hardness of M AX -E3-L IN 2 For any 0 gap(½+ , 1 )-E3-LIN-2 is NP-hard.

26
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA - Folding In order to ensure that A is balanced we force A( f ) = A(f ) by reading only half of A: A(f ) = A(f ) if f(1) = 1 A( f ) if f(1) =

27
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Testing the SWA – L 2 ( ) Håstad ’ s L ABEL C OVER Test E w,v [ Â B 1 2 E w,v [ Â B ^^^^

28
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad x - ½ e -x / 2

29
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad x - ½ e -x / 2 e -x 1 -x

30
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Hardness of M AX -E3-L IN 2 For any 0 it is NP-hard to approximate M AX -E3-L IN -2 within a factor of 2 . M AX -E3-L IN -2 is non-approximable beyond the random assignment threshold.

31
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Overview gap( ,1) L ABEL C OVER gap(½+ , 1 ) E3-LIN-2 gap(⅞+ , 1 ) 3SAT Long Code + Håstad’s L ABEL C OVER Junta testing 3SAT gap(c,1) 3SAT PCP theorem Parallel Repetition Theorem 4-gadget

32
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad Hardness of M AX -E3-S AT For any 0 it is NP-hard to approximate M AX -E3-S AT within a factor of 8/7 .

33
Inapproximability Seminar – 2005 Some Optimal Inapproximability Results – Johan H å stad FIN

Similar presentations

OK

Dana Moshkovitz MIT. Take-home message: Prove hardness results assuming The Projection Games Conjecture (PGC)! Theorem: If SAT requires exponential.

Dana Moshkovitz MIT. Take-home message: Prove hardness results assuming The Projection Games Conjecture (PGC)! Theorem: If SAT requires exponential.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on different means of transport Ppt on earth damage Ppt on success and failure of reconstruction Ppt on inbound tourism in india Convert word doc to ppt online Ppt on conservation of forest in india Ppt on area of trapezoid Ppt on hydrostatic forces on submerged surfaces Ppt on low level language programming Ppt on non biodegradable waste product