1 / 33

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.

claus
Download Presentation

Some Optimal Inapproximability Results

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related