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

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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 –

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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

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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.

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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

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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

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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]

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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.

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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.

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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

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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.

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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

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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

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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}

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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

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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

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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)

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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.

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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

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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).

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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.

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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

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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

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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.

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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.

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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) =

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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 ^^^^

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

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

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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.

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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

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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 .

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

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