Vol.1: Geometry Subhash Khot IAS Elchanan Mossel UC Berkeley Guy Kindler DIMACS Ryan O’Donnell IAS

It is impossible to improve the MAX-CUT approximation of Goemans and Williamson. It is impossible to improve the MAX-CUT approximation of Goemans and Williamson. (assuming two unproven conjectures…): 1.The Unique Games conjecture [Khot02] 2.The “Majority is Stablest” conjecture. We show:

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

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

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

HistoryHistory [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,  GW = min ≈.878 (arccos ρ) / π ½ - ½ ρ -1 < ρ < 1 ½ 1 10 ρ −1 −.69 = : ρ* 87.8%

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

HistoryHistory [Bellare-Goldreich-Sudan ’92] more than 83/84 is NP-hard [Håstad ’97] 16/17  is NP-hard [  GW = > easy > 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

the conjectures Unique Games conjecture:  MAX-2LIN(q) is hard. Input: two-variable linear equations mod q=10 ⁶. 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!

“Beating Goemans-Williamson “Beating Goemans-Williamson – i.e., approximating MAX-CUT to a factor.879 – is formally harder* than the problem of is formally harder* than the problem of satisfying 1% of a given set of 99%-satisfiable two-variable linear equations mod 10 ⁶.” So, Uri Zwick et al, please work on this problem, rather than this problem. How we want you to interpret our result

oProvides insight to Unique Games conjecture. oFourier methods and related results independently interesting. oMotivates algorithmic progress on MAX-2SAT, MAX- 2LIN(q) More motivation for result

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

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

the GW algorithm v G=(V,E): xuxu xuxu u xvxv xvxv (unit sphere in R n )

the GW algorithm v G=(V,E): xuxu xuxu u In S 0, this is Max-Cut! xvxv xvxv

the GW algorithm v xvxv xvxv G=(V,E): xuxu xuxu u

GW algorithm: performance x v  S n-1 xuxu xuxu xvxv xvxv xuxu xuxu xuxu xvxv arccos( ) donation to (*) Pr[(x u,x v ) is cut]= arccos( )/  arccos( )/ 

GW algorithm: performance Pr[(x u,x v ) is cut]= arccos( )/  arccos( )/  xuxu xvxv arccos( ) donation to (*) Actually this is tight.. Tight, if all inner products are ρ*

Important example: G ρ V = S n-1 E = {(x,y) :  ρ} [FS] a hyperplane cut is optimal for G ρ size of cut: (arccos ρ)/π ρ - negative

More important example: D ρ V = {-1,1} n  S n-1 a random edge (x,y): x~{-1,1} n, w(x,y) = P[(x,y) is chosen] w(x,y) = P[(x,y) is chosen] E[ ] = ρ higher probability tightly concentrated well, actually {-n -½,n ½ } n y: y i = x i w.p. ½ + ½ρ -x i w.p. ½ - ½ρ

OPT(D ρ ) = OPT(G ρ ) = (arccos ρ)/π? no… For f(x) = x 7, w( f -1 (1),f -1 (-1) ) = P[x 7 ≠ y 7 ] = ½ - ½ ρ = ½ - ½ ρ For f(x) = sign(  x i ) = Maj(x), w( f -1 (1),f(-1) )=P[Maj(x)≠Maj(y)] ≈ (arccos ρ)/π ≈ (arccos ρ)/π Are D ρ and G ρ similar? Dictatorship not at all a dictatorship

Do D ρ and G ρ act the same?  non dictatorship: f : {-1,1} n ―> {-1,1} s.t. for all n dictatorships, “correlation” with f is at most  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) +o  (1)

A dictatorship test the test: f : {-1,1} n ―> {-1,1}, pick x,y as before, pick x,y as before, verify that f(x)≠f(y). 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) +o  (1)

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

Guy Kindler DIMACS Ryan O’Donnell IAS Guy Kindler DIMACS Ryan O’Donnell IAS Subhash Khot IAS Elchanan Mossel UC Berkeley Subhash Khot IAS Elchanan Mossel UC Berkeley Vol.2: Main results

Unique Games Conjecture “Unique Label Cover” with q colors: n Labels [q] π uv π uv : Bijections Input π uv

Unique Games Conjecture “Unique Label Cover” with q colors: n Labels [q] π uv π uv : Bijections Assignment π uv

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

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

Main theorem – proof overview π uv Assignment --> Cut fufu fvfv Max-Cut Test: Verify f v (x)≠f u (y)

Main theorem – proof overview π uv Assignment --> Cut fufu fvfv Max-Cut Test: Verify f v (x)≠f u (y) Max-Cut Test: Verify f v (x)≠f u ( π uv ( y)) x π uv (y) Completeness:  at least (1-  )(1-ρ)/2 Soundness:  at most (1-  `)(arccos ρ)/ 

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.

QuestionsQuestions 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 o q (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.