Vol.1: Geometry

1. Hardness of approximating Vol.1: Geometry MAX-CUT Subhash KhotIAS Elchanan MosselUC Berkeley Guy KindlerDIMACS Ryan O’DonnellIAS

2. We show: • It is impossible to improve the MAX-CUT approximation of Goemans and Williamson. • (assuming two unproven conjectures…): • The Unique Games conjecture [Khot02] • The “Majority is Stablest” conjecture.

3. Conjectures? What? • Usual modus operandi in Mathematics: • Prove theorem, give talk. • Non-usual modus operandi in Mathematics: • Fail to prove two theorems, give talk.

4. What is MAX-CUT? • G = (V,E) • C = (S,S), partition of V • w(C) = |(SxS)  E| • w : E ―> R+ weighted  unweighted

5. What is MAX-CUT? • OPT = OPT(G) = maxc {|C|} • MAX-CUT problem:find C with w(C)= OPT • -approximation:find C with w(C) ≥ ·OPT

6. 1 ½ 87.8% (arccos ρ) / π GW = min ≈ .878 ρ ½ - ½ ρ -1 < ρ < 1 0 1 −.69 =:ρ* −1 History [Karp ’72] MAX-CUT is NP-complete. [Shani-Gonzalez ’76] ½-approximation (partition vertices randomly) [’76-’94] no progress… (½+o(1) approx.) [Goemans-Williamson ’94]GW-approximation,

7. (arccos ρ) / π GW = min ≈ .878 ½ - ½ ρ -1 < ρ < 1 History Intrinsic? Coincidence? [Goemans-Williamson ’94]GW-approximation, G = (V,E) ―> geometric problem ―> random cut

8. History • [Bellare-Goldreich-Sudan ’92] more than 83/84 is NP-hard • [Håstad ’97] 16/17  0.941 is NP-hard • [ GW=0.878 -> easy 0.941 -> hard ] • other results: • [Karloff ’99, Feige-Schechtman ’99]GW does not perform any better than GW. • [Alon Sudakov ’98]Same holds even for the discrete cube

9. the conjectures • Unique Games conjecture: MAX-2LIN(q) is hard. • Input:two-variable linear equations mod q=10⁶. You know that 99% can be satisfied. • Goal: satisfy 1%. • status: MAX-2LIN(2) is hard for some parameters… • Majority is Stablest conjecture: among balanced f:{1,-1}n{1,-1}, where each coordinate has “small influence,” the Majority function is least sensitive to noise. • status: everybody knows it’s true!

10. How we want you tointerpret our result • “Beating Goemans-Williamson • – i.e., approximating MAX-CUT to a factor .879 – • is formally harder* than the problem of • satisfying 1% of a given set of 99%-satisfiable two-variable linear equations mod 10⁶.” • So, Uri Zwick et al, • please work on this problem, • rather than this problem.

11. More motivation for result • Provides insight to Unique Games conjecture. • Fourier methods and related results independently interesting. • Motivates algorithmic progress on MAX-2SAT, MAX-2LIN(q)

12. What’s next in this talk • “Maj is Stablest”  long-code test with soundness/completeness=GW ,and the relation to the geometry in GW algorithm. • and if times permits: • Hardness for MAX-CUT, from Unique Games conjecture + long-code test • Discussion of “Maj is Stablest” and partial results. • Discussion of Unique Games conjecture.

13. The long-code • Encodes elements in {1,2,..,q} • The encoding of 2{1,2,3}: • 1 1 1 11 1 -1 11 -1 1 -11 -1 -1 -11 -1 -1 1… .. • In general, i{1,..,q} is encoded by f:{1,-1}q{-1,1}, • defined by f(x)=xi

14. the GW algorithm G=(V,E): v xv xu u (unit sphere in Rn)

15. the GW algorithm In S0, this is Max-Cut! G=(V,E): v xv u xu

16. the GW algorithm G=(V,E): v xv xu u

17. xv xv xvSn-1 xu xu <xu,xv> xu donation to (*) GW algorithm: performance Pr[(xu,xv) is cut]= arccos(<xu,xv>)/ arccos(<xu,xv>)

18. xv <xu,xv> arccos(<xu,xv>) xu donation to (*) GW algorithm: performance Actually this is tight.. Pr[(xu,xv) is cut]= arccos(<xu,xv>)/ Tight, if all inner products are ρ*

19. Important example: Gρ ρ - negative • V = Sn-1 • E = {(x,y) : <x,y>  ρ} • [FS] a hyperplane cut is optimal for Gρ • size of cut: (arccos ρ)/π

20. xi w.p. ½ + ½ρ -xi w.p. ½ - ½ρ y: yi = More important example: Dρ well, actually {-n-½,n½}n • V = {-1,1}n Sn-1 • a random edge (x,y): x~{-1,1}n, • w(x,y) = P[(x,y) is chosen] • E[<x,y>] = ρ higher probability tightly concentrated

21. Are Dρ and Gρ similar? • OPT(Dρ) = OPT(Gρ) = (arccos ρ)/π? • no… • For f(x) = x7, • w( f-1(1),f-1(-1) ) = P[x7 ≠ y7] • = ½ - ½ ρ • For f(x) = sign(xi) = Maj(x), • w( f-1(1),f(-1) )=P[Maj(x)≠Maj(y)] • ≈ (arccos ρ)/π Dictatorship not at all a dictatorship

22. Do Dρ and Gρ act the same? •  non dictatorship: • f : {-1,1}n ―> {-1,1} s.t. • for all n dictatorships, “correlation” with f is at most  • conjecture: • If f is  non dictatorship, • w( (f-1(1),f-1 (-1) )  (arccos ρ)/π • +o(1)

23. A dictatorship test • the test: f : {-1,1}n ―> {-1,1}, • pick x,y as before, • verify that f(x)≠f(y). • dictatorships: pass w.p. ½ - ½ρ • non-dictatorships: pass w.p. • conjecture: • If f is  non dictatorship, • w( (f-1(1),f(-1) )  (arccos ρ)/π • +o(1)

24. A dictatorship test • the test: f : {-1,1}n ―> {-1,1}, • pick x,y as before, • verify that f(x)≠f(y). • dictatorships: pass w.p. ½ - ½ρ • non-dictatorships: pass w.p. •  (arccos ρ)/π • . a long-code test long-code words completeness soundness gap: (arccos ρ)/π ½ - ½ρ soundness completeness = ≈ .878 (for ρ = ρ*)

26. Hardness of approximating MAX-CUT Vol.2: Main results Subhash KhotIAS Elchanan MosselUC Berkeley Guy KindlerDIMACS Ryan O’DonnellIAS

27. πuv πuv πuv πuv n πuv πuv Bijections πuv πuv πuv πuv: πuv Unique Games Conjecture • “Unique Label Cover” with q colors: Labels [q] πuv Input

28. πuv πuv πuv πuv n πuv πuv Bijections πuv πuv πuv πuv: πuv Unique Games Conjecture • “Unique Label Cover” with q colors: Labels [q] πuv Assignment

29. πuv πuv πuv πuv n πuv πuv Bijections πuv πuv πuv πuv: πuv Unique Games Conjecture • Conjecture: satisfying fraction of edges is hard, even if 1- of them can be satisfied. Labels [q] πuv Assignment

30. Unique Games Conjecture • Conjecture: satisfying fraction of edges is hard, even if 1- of them can be satisfied. • UGC is stronger than AS+ALMSS+Raz altogether. • UGC implies • MIN-2SAT-Deletion hard to approximate to within any constant factor [Hastad, Khot ’02] • Vertex-Cover hard to approximate to within any factor smaller than 2 [Khot-Regev ’03] • These results need long-code tests, relying on theorems in Fourier analysis. [Bourgain ’02; Friedgut ’98]

31. πuv Main theorem – proof overview Assignment --> Cut fv fu Max-Cut Test: Verify fv(x)≠fu(y)

32. πuv Main theorem – proof overview Assignment --> Cut x fv πuv (y) fu Max-Cut Test: Verify fv(x)≠fu(πuv(y)) Max-Cut Test: Verify fv(x)≠fu(y) Completeness: at least (1-)(1-ρ)/2 Soundness: at most (1-`)(arccos ρ)/

33. More results • thm: “Majority is Stablest” holds for threshold functions. • thm: among balanced functions where each coordinate has small influence, Majority has the most weight on level 1. • corr: Assuming UGC alone, MAX-CUT is hard to approx. to within .909 < 16/17 = .941 • thm: Assuming UGC, MAX-2LIN(q) is hard to approx. to within any constant factor.

34. Questions • Prove Majority Is Stablest Conjecture. • What balanced q-ary function f : [q]ⁿ [q] is stablest? Plurality? • Thm [us]: Noise stability of Plurality is q(ρ-1)/(ρ+1) + o(1). • If q-ary stability is oq(1), then UGC implies hardness of (hence, essentially, equivalence with) MAX-2LIN(q). • A sharp bound for the q-ary stability problem would give strong results for the UGC w.r.t. how big q needs to be as a function of ε.