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

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 /11 –  E2-SAT /21 –  Max-Cut /16 –  Max-di-Cut /11 –  Vertex Cover2 7/6 – 

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

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.

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

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

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]

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.

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.

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

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.

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

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

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}

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

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

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

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.

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 

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).

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.

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 

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 

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.

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.

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) = 

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           ^^^^

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

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

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.

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

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  .

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