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Some Optimal Inapproximability Results

Some Optimal Inapproximability Results. Johan Hå st ad Royal Institute of Technology, Sweden 2002. Bound Summary. 3SAT. gap ( c ,1) 3SAT. PCP theorem. Parallel Repetition Theorem. 4-gadget. Overview. gap( ⅞ + e , 1 - e ) 3SAT. Long Code + H å stad’s L ABEL C OVER Junta testing.

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Some Optimal Inapproximability Results

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  1. Some Optimal Inapproximability Results Johan Håstad Royal Institute of Technology, Sweden 2002

  2. Bound Summary Some Optimal Inapproximability Results – Johan Håstad

  3. 3SAT gap(c,1) 3SAT PCP theorem ParallelRepetitionTheorem 4-gadget Overview gap(⅞+e, 1-e) 3SAT Long Code + Håstad’s LABELCOVER Junta testing gap(½+e, 1-e) E3-LIN-2 gap(e,1) LABELCOVER Some Optimal Inapproximability Results – Johan Håstad

  4. Hardness of MAX-E3-SAT gap(½+e, 1-e)-E3-LIN-2 can be reduced togap(⅞+¼e, 1-¼e)-E3-SAT. Some Optimal Inapproximability Results – Johan Håstad

  5. (xVyVz),(xVyVz),(xVyVz),(xVyVz) (xVyVz),(xVyVz),(xVyVz),(xVyVz) 4-gadget Hardness of MAX-E3-SAT • xyz = 1 • xyz = -1 gap(½+e, 1-e)-E3-LIN-2 can be reduced togap(⅞+¼e, 1-¼e)-E3-SAT. Some Optimal Inapproximability Results – Johan Håstad

  6. 3SAT gap(c,1) 3SAT PCP theorem ParallelRepetitionTheorem 4-gadget Overview gap(⅞+e, 1-e) 3SAT Long Code + Håstad’s LABELCOVER Junta testing gap(½+e, 1-e) E3-LIN-2 gap(e,1) LABELCOVER Some Optimal Inapproximability Results – Johan Håstad

  7. LABEL COVER • An instance of the LABEL COVER problem is denoted by:L(G(V,W,E) ,[n] ,[m] ,P) where: • G(V,W,E) is a regular bipartite graph. • [n], [m] are sets of labels for V, W. • P = {pwv}(v,w)EFor every edge (v,w) pwv is a map pwv:[m][n] Some Optimal Inapproximability Results – Johan Håstad

  8. LABEL COVER • A labeling s:V[n], W[m]satisfies pwv if pwv(s (w)) = s (v). • For an instance L, The maximum fraction of constraints pwv that can be satisfied by any labeling is denoted by OPT(L). • The goal: Find a labeling s that satisfies OPT(L) of the constraints. Some Optimal Inapproximability Results – Johan Håstad

  9. 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] ,P)is NP-hard. Some Optimal Inapproximability Results – Johan Håstad

  10. LABEL COVER - Repetition • Given L(G(V,W,E) ,[n] ,[m] ,P)define Lk(G(V,W,E) ,[n] ,[m] ,P) : • V:= Vk W:= Wk • [n] := [n]k [m] := [m]k • (v,w)Efor v=(v1,…,vk) w=(w1,…,wk) iff i[k] (vi,wi)E • For every pwvPdefine:pwv(m1,…,mk) = (pw1v1(m1),…,pwkvk(mk)) V :=Vk [n] := [n]k Some Optimal Inapproximability Results – Johan Håstad

  11. 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) cck . Some Optimal Inapproximability Results – Johan Håstad

  12. LABEL COVER - Conclusion • For every t> 0 there are Nt, Mt s.t.the gap-LABEL COVER problem:gap(t,1)-L(G(V,W,E) ,[Nt] ,[Mt] ,P)is NP-hard Some Optimal Inapproximability Results – Johan Håstad

  13. 3SAT gap(c,1) 3SAT PCP theorem ParallelRepetitionTheorem 4-gadget Overview gap(⅞+e, 1-e) 3SAT Long Code + Håstad’s LABELCOVER Junta testing gap(½+e, 1-e) E3-LIN-2 gap(e,1) LABELCOVER Some Optimal Inapproximability Results – Johan Håstad

  14. 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

  15. Fourier Analysis - Reminder • Linear functions: a[n] Xa(x):= Piaxi • Inner Product Space:<A,B>:= Ex[A(x)B(x)] • <Xa,Xb> = dab • {Xa}a[n] is an orthonormal basis for {±1}[n]R Some Optimal Inapproximability Results – Johan Håstad

  16. Fourier Analysis - Reminder • Every A:{±1}[n]  {±1}can be written as: A = a[n]ÂaXa • {Âa}a[n] are called the Fourier coefficients of A. • Parseval’s identity:for any boolean function A we havea[n]Âa2 = 1 Some Optimal Inapproximability Results – Johan Håstad

  17. Fourier Analysis - Reminder • Âa= <A,Xa> • Prx[A(x) = Xa(x)] = ½ + ½Âa • Â= Ex[A(x)] • X{i}(x) = xi = LCi(x)(Dictatorship) Some Optimal Inapproximability Results – Johan Håstad

  18. Testing the Long CodeLinearity Test • Choose f,g{±1}[n] at random. • Check if:A(f)A(g) = A(fg) • Perfect completeness. Some Optimal Inapproximability Results – Johan Håstad

  19. -1 with probability e 1 with probability 1-e Testing the Long CodeJunta Test, parameterized by e • Choose f,g{±1}[n] at random. • Choose m{±1}[n] by setting:x[n] m(x) = • Check if:A(f)A(g) = A(fgm) Some Optimal Inapproximability Results – Johan Håstad

  20. Standard Written Assignmentfor the LABEL COVER • Given a LABEL COVER problemL(G(V,W,E) ,[n] ,[m] ,P)And an assignments that satisfy all the constraints, • The SWA(s ) contains for every vV theLong Code of it’s assignment LCs(v)and for every wW it’s LCs(w). Some Optimal Inapproximability Results – Johan Håstad

  21. Testing the SWA – L2(e)Håstad’s LABEL COVERTest • Given: • LABEL COVER problem L(G(V,W,E) ,[n] ,[m] ,P) • A supposed SWA for it. • Choose (v,w)Eat random. • Denote (the supposed) LCs(v)by A and (the supposed) LCs(w)by B. Some Optimal Inapproximability Results – Johan Håstad

  22. -1 with probability e 1 with probability 1-e Testing the SWA – L2(e)Håstad’s LABEL COVERTest • Choose f{±1}[n]at random. • Choose g{±1}[m]at random. • Choose m{±1}[m] by setting:x[m] m(x) = • Check if:A(f)B(g) = B((fopwv)gm) Some Optimal Inapproximability Results – Johan Håstad

  23. Testing the SWA – L2(e)Håstad’s LABEL COVERTest • Completeness: 1-e Some Optimal Inapproximability Results – Johan Håstad

  24. Testing the SWA – L2(e)Håstad’s LABEL COVERTest • Completeness: 1-e • Soundness: • For any LABEL COVER problem Land any e,d >0, if the probability thattest L2(e) accepts is ½(1+d) thenthere is a assignment s that satisfy 4ed2of L`s constraints. Some Optimal Inapproximability Results – Johan Håstad

  25. Hardness of MAX-E3-LIN2 For any e>0gap(½+e, 1-e)-E3-LIN-2 is NP-hard. Some Optimal Inapproximability Results – Johan Håstad

  26. A(f) if f(1) = 1 -A(-f) if f(1) = -1 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) = Some Optimal Inapproximability Results – Johan Håstad

  27. Testing the SWA – L2(e)Håstad’s LABEL COVERTest Ew,v[bÂp(b)Bb2(1-2e)|b|] = d Ew,v[bÂ2p(b)Bb2 |b|-1]  4ed2 ^ ^ Some Optimal Inapproximability Results – Johan Håstad

  28. x-½e-x/2 Some Optimal Inapproximability Results – Johan Håstad

  29. e-x 1-x x-½e-x/2 Some Optimal Inapproximability Results – Johan Håstad

  30. Hardness of MAX-E3-LIN2 • For any e>0 it is NP-hard to approximateMAX-E3-LIN-2 within a factor of 2-e. • MAX-E3-LIN-2 is non-approximable beyond the random assignment threshold. Some Optimal Inapproximability Results – Johan Håstad

  31. 3SAT gap(c,1) 3SAT PCP theorem ParallelRepetitionTheorem 4-gadget Overview gap(⅞+e, 1-e) 3SAT Long Code + Håstad’s LABELCOVER Junta testing gap(½+e, 1-e) E3-LIN-2 gap(e,1) LABELCOVER Some Optimal Inapproximability Results – Johan Håstad

  32. Hardness of MAX-E3-SAT For any e>0 it is NP-hard to approximateMAX-E3-SAT within a factor of 8/7-e. Some Optimal Inapproximability Results – Johan Håstad

  33. FIN Some Optimal Inapproximability Results – Johan Håstad

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