Vol.1: GeometryVol.1: GeometrySubhash Khot

IAS

Elchanan MosselUC Berkeley

Guy KindlerDIMACS

Ryan O’DonnellIAS

It is impossible to improve the MAX-CUT It is impossible to improve the MAX-CUT

approximation of Goemans and Williamson.approximation of Goemans and Williamson.

(assuming two unproven conjectures…):(assuming two unproven conjectures…):

1.1. The Unique Games conjecture The Unique Games conjecture [Khot02][Khot02]

2.2. The “Majority is Stablest” conjecture.The “Majority is Stablest” conjecture.

We show:We show:We show:We show:

Conjectures? What?Conjectures? What?Conjectures? What?Conjectures? What?

Usual modus operandi in Mathematics:Usual modus operandi in Mathematics:

Prove theorem, give talk.Prove theorem, give talk.

Non-usual modus operandi in Mathematics:Non-usual modus operandi in Mathematics:

Fail to prove two theorems, give talk.Fail to prove two theorems, give talk.

What is MAX-CUT?What is MAX-CUT?What is MAX-CUT?What is MAX-CUT?

G = (V,E)G = (V,E)

C = (S,S), partition of VC = (S,S), partition of V

w(C) = |(SxS) w(C) = |(SxS) E| E|

w : E ―> Rw : E ―> R++

weighted unweightede (V1 V2) E

w(C) w(e)

What is MAX-CUT?What is MAX-CUT?What is MAX-CUT?What is MAX-CUT?

OPT = OPT(G) = maxOPT = OPT(G) = maxc c {|C|}{|C|}

MAX-CUT problem:MAX-CUT problem:

find C with w(C)= OPTfind C with w(C)= OPT

-approximation:-approximation:

find C with w(C) ≥ find C with w(C) ≥ ·OPT ·OPT

HistoryHistoryHistoryHistory

[Karp ’72][Karp ’72] MAX-CUT is NP-complete. MAX-CUT is NP-complete.

[Shani-Gonzalez ’76][Shani-Gonzalez ’76] ½-approximation ½-approximation(partition vertices randomly)(partition vertices randomly)

[’76-’94][’76-’94] no progress… (½+o(1) approx.) no progress… (½+o(1) approx.)

[Goemans-Williamson ’94][Goemans-Williamson ’94] GWGW-approximation,-approximation,

GWGW = min = min ≈ .878≈ .878

(arccos (arccos ρρ) / ) / ππ

½ - ½ ½ - ½ ρρ-1 < -1 < ρρ < < 11

½

1

10ρ

−1

−.69 =: ρ*

87.8%

HistoryHistoryHistoryHistory

[Goemans-Williamson ’94][Goemans-Williamson ’94] GWGW-approximation,-approximation,

GWGW = min = min ≈ .878≈ .878

(arccos (arccos ρρ) / ) / ππ

½ - ½ ½ - ½ ρρ-1 < -1 < ρρ < < 11

G = (V,E)G = (V,E)

― ―> geometric problem> geometric problem

― ―> random > random cutcut

Intrinsic? Coincidence?

HistoryHistoryHistoryHistory

[Bellare-Goldreich-Sudan ’92] [Bellare-Goldreich-Sudan ’92] more than 83/84 is NP-hard more than 83/84 is NP-hard

[Håstad ’97] [Håstad ’97] 16/17 16/17 0.941 is NP-hard 0.941 is NP-hard

[ [ GWGW=0.878 -> easy 0.941 -> hard ]=0.878 -> easy 0.941 -> hard ]

other results:other results:

[Karloff ’99, Feige-Schechtman ’99][Karloff ’99, Feige-Schechtman ’99]

GW does not perform any better than GW does not perform any better than GWGW..

[Alon Sudakov ’98][Alon Sudakov ’98]

Same holds even for the discrete cubeSame holds even for the discrete cube

the conjecturesthe conjecturesthe conjecturesthe conjectures

Unique Games conjecture:Unique Games conjecture: MAX-2LIN(q) is hard. MAX-2LIN(q) is hard.

Input:Input: twotwo-variable linear equations mod q=10⁶. -variable linear equations mod q=10⁶. You You knowknow that 99% can be satisfied. that 99% can be satisfied.

Goal:Goal: satisfy 1%. satisfy 1%.

status:status: MAX-2LIN(2) is hard for MAX-2LIN(2) is hard for somesome parameters… parameters…

Majority is Stablest conjecture:Majority is Stablest conjecture: among balanced f:{1,- among balanced f:{1,-1}1}nn{1,-1}, where each coordinate has “small {1,-1}, where each coordinate has “small influence,” influence,” the Majority function is least sensitive to noise. the Majority function is least sensitive to noise.

status:status: everybody knows it’s true! everybody knows it’s true!

““Beating Goemans-WilliamsonBeating Goemans-Williamson

– – i.e., approximating MAX-CUT to a factor .879 –i.e., approximating MAX-CUT to a factor .879 –

is formally harder* than the problem ofis formally harder* than the problem of

satisfying 1% of a given set of 99%-satisfiable satisfying 1% of a given set of 99%-satisfiable two-variable linear equations mod 10⁶.”two-variable linear equations mod 10⁶.”

So, Uri Zwick So, Uri Zwick et alet al, ,

please work on this problem,please work on this problem,

rather than this problem.rather than this problem.

How we want you toHow we want you tointerpret our resultinterpret our resultHow we want you toHow we want you tointerpret our resultinterpret our result

o Provides insight to Unique Games conjecture.Provides insight to Unique Games conjecture.

o Fourier methods and related results Fourier methods and related results independently interesting.independently interesting.

o Motivates algorithmic progress on MAX-2SAT, Motivates algorithmic progress on MAX-2SAT, MAX-2LIN(q)MAX-2LIN(q)

More motivation for resultMore motivation for resultMore motivation for resultMore motivation for result

What’s next in this talkWhat’s next in this talkWhat’s next in this talkWhat’s next in this talk

o ““Maj is Stablest” Maj is Stablest” long-code test with long-code test with

soundness/completeness=soundness/completeness=GW GW ,,

and the relation to the geometry in GW algorithm.and the relation to the geometry in GW algorithm.

and if times permits:and if times permits:

o Hardness for MAX-CUT, from Unique Games Hardness for MAX-CUT, from Unique Games

conjecture + long-code testconjecture + long-code test

o Discussion of “Maj is Stablest” and partial results. Discussion of “Maj is Stablest” and partial results.

o Discussion of Unique Games conjecture. Discussion of Unique Games conjecture.

The long-codeThe long-codeThe long-codeThe long-code

Encodes elements in {1,2,..,q}Encodes elements in {1,2,..,q}

The encoding of 2The encoding of 2{1,2,3}:{1,2,3}:1 1 11 1 1 111 1 -11 1 -1 111 -1 11 -1 1 -1-11 -1 -11 -1 -1 -1-11 -1 -11 -1 -1 11…… ....

In general, iIn general, i{1,..,q} is encoded by f:{1,-1}{1,..,q} is encoded by f:{1,-1}qq{-{-

1,1},1,1},

defined by f(x)=xdefined by f(x)=xii

the GW algorithmthe GW algorithmthe GW algorithmthe GW algorithm

u v

(u,v) E

maximize

1- x ,x (*)

2

vv

G=(V,E):G=(V,E):

xuxuuuxvxv

2

u v

(u,v) E

x x=

4

(unit sphere in R(unit sphere in Rnn))

the GW algorithmthe GW algorithmthe GW algorithmthe GW algorithm

u v

(u,v) E

maximize

1- x ,x (*)

2

vv

G=(V,E):G=(V,E):

xuxu

uu

opt

In S0, this is Max-Cut!

xvxv

the GW algorithmthe GW algorithmthe GW algorithmthe GW algorithm

u v

(u,v) E

maximize

1- x ,x (*)

2

vv xvxv

G=(V,E):G=(V,E):

xuxuuu

GW algorithm: performanceGW algorithm: performanceGW algorithm: performanceGW algorithm: performance

u v

(u,v) E

1- x ,x(*)

2

xvSn-1xvSn-1

xuxu

xvxv xuxu

xu

xv

arccos(<xarccos(<xuu,x,xvv>)>)arccos(<xarccos(<xuu,x,xvv>)>)

donation to (*)donation to (*)donation to (*)donation to (*)

<x<xuu,x,xvv>><x<xuu,x,xvv>>

Pr[(xPr[(xuu,x,xvv) is cut]=) is cut]=

arccos(<xarccos(<xuu,x,xvv>)/>)/

u v(u,v) E

E[w(cut)]

arccos( x ,x )/

GW algorithm: performanceGW algorithm: performanceGW algorithm: performanceGW algorithm: performance

u v

(u,v) E

1- x ,x(*)

2

Pr[(xPr[(xuu,x,xvv) is cut]=) is cut]=

arccos(<xarccos(<xuu,x,xvv>)/>)/

u v(u,v) E

E[w(cut)]

arccos( x ,x )/

xu

xv

arccos(<xarccos(<xuu,x,xvv>)>)arccos(<xarccos(<xuu,x,xvv>)>)

donation to (*)donation to (*)donation to (*)donation to (*)

<x<xuu,x,xvv>><x<xuu,x,xvv>>GW

arccos( ) /min

(1 )/ 2

Actually this is tight..

Tight, if all inner products are ρ*

0.879.. (f or some )

Important example: GImportant example: GρρImportant example: GImportant example: Gρρ

V = SV = Sn-1n-1

E = {(x,y) : E = {(x,y) : <<x,yx,y> > ρρ}}

[FS][FS] a hyperplane cut a hyperplane cut

is optimal for Gis optimal for Gρρ

size of cut: size of cut: (arccos (arccos ρρ)/)/ππ

ρ - negative

More important example: More important example: DDρρ

More important example: More important example: DDρρ

V = {-1,1}V = {-1,1}nn S Sn-1n-1

a random edge (x,y): a random edge (x,y): x~{-1,1} x~{-1,1}nn,,

w(x,y) = P[(x,y) is chosen]w(x,y) = P[(x,y) is chosen]

E[E[<<x,yx,y>>] = ] = ρρ

higher probability

tightly concentrated

well, actually {-n-

½,n½}n

y: yy: yi i = = xxii w.p. ½ + ½ w.p. ½ + ½ρρ

-x-xii w.p. ½ - ½ w.p. ½ - ½ρρ

OPT(DOPT(Dρρ) = OPT(G) = OPT(Gρρ) = (arccos ρ)/) = (arccos ρ)/ππ??

no…no…

For f(x) = xFor f(x) = x77,,

w( fw( f-1-1(1),f(1),f-1-1(-1) ) = P[x(-1) ) = P[x7 7 ≠ y≠ y77]]

= ½ - ½ ρ = ½ - ½ ρ

For f(x) = sign(For f(x) = sign(xxii) = Maj(x),) = Maj(x),

w( fw( f-1-1(1),f(-1) )=P[Maj(x)≠Maj(y)](1),f(-1) )=P[Maj(x)≠Maj(y)]

≈ ≈ (arccos ρ)/(arccos ρ)/ππ

Are DAre Dρρ and G and Gρρ similar? similar?Are DAre Dρρ and G and Gρρ similar? similar?

Dictatorship

not at all a dictatorship

Do DDo Dρρ and G and Gρρ act the same? act the same?Do DDo Dρρ and G and Gρρ act the same? act the same?

non dictatorship: non dictatorship:

f : {-1,1}f : {-1,1}n n ―> {-1,1} s.t.―> {-1,1} s.t. for all n dictatorships,for all n dictatorships, “correlation” with f is “correlation” with f is at most at most

conjecture:conjecture:

If f is If f is non dictatorship, non dictatorship,

w( (fw( (f-1-1(1),f(1),f-1-1 (-1) ) (-1) ) (arccos ρ)/ (arccos ρ)/ππ

+o+o(1)(1)

A dictatorship testA dictatorship testA dictatorship testA dictatorship test

the test: the test: f : {-1,1}f : {-1,1}n n ―> {-1,1},―> {-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:dictatorships: pass w.p. ½ - ½ pass w.p. ½ - ½ρρ

non-dictatorships:non-dictatorships: pass w.p. pass w.p.

conjecture:conjecture:

If f is If f is non dictatorship, non dictatorship,

w( (fw( (f-1-1(1),f(-1) ) (1),f(-1) ) (arccos ρ)/ (arccos ρ)/ππ

+o+o(1)(1)

A dictatorship testA dictatorship testA dictatorship testA dictatorship test

the test: the test: f : {-1,1}f : {-1,1}n n ―> {-1,1},―> {-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:dictatorships: pass w.p. ½ - ½ pass w.p. ½ - ½ρρ

non-dictatorships:non-dictatorships: pass w.p. pass w.p.

(arccos ρ)/(arccos ρ)/ππ

..

long-code words

a long-code test

completeness

soundnessgap:gap:

soundnesssoundness

completenesscompleteness

(arccos ρ)/(arccos ρ)/ππ

½½ - ½ρ - ½ρ≈ ≈ .878 (for ρ = .878 (for ρ =

ρ*)ρ*)==

Guy KindlerDIMACS

Ryan O’DonnellIAS

Guy KindlerDIMACS

Ryan O’DonnellIAS

Subhash KhotIAS

Elchanan MosselUC Berkeley

Subhash KhotIAS

Elchanan MosselUC Berkeley

Vol.2: Main resultsVol.2: Main results

Unique Games ConjectureUnique Games ConjectureUnique Games ConjectureUnique Games Conjecture““Unique Label Cover” with Unique Label Cover” with qq colors: colors:

n

Labels

[q]

πu

v

πuv :

Bijections

Input

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

Unique Games ConjectureUnique Games ConjectureUnique Games ConjectureUnique Games Conjecture““Unique Label Cover” with Unique Label Cover” with qq colors: colors:

n

Labels

[q]

πu

v

πuv :

Bijections

Assignment

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

Unique Games ConjectureUnique Games ConjectureUnique Games ConjectureUnique Games Conjecture

n

Labels

[q]

πu

v

πuv :

Bijections

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

πuv

Conjecture:Conjecture: satisfying satisfying fraction of edges is hard, even fraction of edges is hard, even

if 1-if 1- of them can be satisfied. of them can be satisfied.

Assignment

Unique Games ConjectureUnique Games ConjectureUnique Games ConjectureUnique Games ConjectureConjecture:Conjecture: satisfying satisfying fraction of edges is hard, even if fraction of edges is hard, even if

1-1- of them can be satisfied. of them can be satisfied.

o UGC is stronger than AS+ALMSS+Raz altogether. UGC is stronger than AS+ALMSS+Raz altogether.

UGC impliesUGC implies

o MIN-2SAT-Deletion hard to approximate to within MIN-2SAT-Deletion hard to approximate to within

any constant factor any constant factor [Hastad, Khot ’02][Hastad, Khot ’02]

o Vertex-Cover hard to approximate to within any Vertex-Cover hard to approximate to within any

factor smaller than 2 factor smaller than 2 [Khot-Regev ’03][Khot-Regev ’03]

o These results need long-code tests, relying on These results need long-code tests, relying on

theorems in Fourier analysis. theorems in Fourier analysis. [Bourgain ’02; Friedgut ’98] [Bourgain ’02; Friedgut ’98]

Main theorem – proof Main theorem – proof overviewoverview

Main theorem – proof Main theorem – proof overviewoverview

πuv

Assignment --> Cut

fu

fv

Max-Cut Test: Verify fv(x)≠fu(y)

Main theorem – proof Main theorem – proof overviewoverview

Main theorem – proof Main theorem – proof overviewoverview

πuv

Assignment --> Cut

fu

fv

Max-Cut Test: Verify fv(x)≠fu(y)Max-Cut Test: Verify fv(x)≠fu(πuv

(y))

x

πuv (y)

Completeness: at least (1-)(1-ρ)/2

Soundness: at most (1-`)(arccos ρ)/

More resultsMore resultsMore resultsMore results

thm:thm: “Majority is Stablest” holds for threshold functions. “Majority is Stablest” holds for threshold functions.

thm:thm: among balanced functions where each coordinate among balanced functions where each coordinate

has small influence, Majority has the most weight on has small influence, Majority has the most weight on

level 1.level 1.

corr:corr: Assuming UGC alone, MAX-CUT is hard to approx. Assuming UGC alone, MAX-CUT is hard to approx.

to within .909 < 16/17 = .941to within .909 < 16/17 = .941

thm:thm: Assuming UGC, MAX-2LIN(q) is hard to approx. to Assuming UGC, MAX-2LIN(q) is hard to approx. to

within any constant factor.within any constant factor.

QuestionsQuestionsQuestionsQuestions

Prove Majority Is Stablest Conjecture.Prove Majority Is Stablest Conjecture.

What balanced What balanced qq-ary function -ary function ff : [ : [qq]]ⁿ ⁿ [ [qq] is stablest? ] is stablest?

Plurality?Plurality?

Thm [us]:Thm [us]: Noise stability of Plurality is q Noise stability of Plurality is q((ρρ-1)/(-1)/(ρρ++1) + o(1)1) + o(1). .

If q-ary stability is If q-ary stability is ooqq(1), then UGC implies hardness (1), then UGC implies hardness

of (hence, essentially, equivalence with) MAX-2LIN(q).of (hence, essentially, equivalence with) MAX-2LIN(q).

A sharp bound for the A sharp bound for the qq-ary stability problem would -ary stability problem would

give strong results for the UGC w.r.t. how big give strong results for the UGC w.r.t. how big qq needs needs

to be as a function of to be as a function of εε..