Transcript Optimal Inapproximability Results for MAX-CUT and Other 2-variable CSPs?

Vol.1: Geometry
Subhash Khot
IAS
Guy Kindler
DIMACS
Elchanan Mossel
UC Berkeley
Ryan O’Donnell
IAS
We show:
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.
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+
w(C) 

e(V1V2) E
w(e)
weighted  unweighted
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
History
[Karp ’72]
MAX-CUT is NP-complete.
½-approximation
(partition vertices randomly)
[Shani-Gonzalez ’76]
[’76-’94]
1
½
87.8%
no progress… (½+o(1) approx.)
[Goemans-Williamson ’94]
ρ
−1
−.69 =: ρ*
0
GW-approximation,
GW = min
-1 < ρ < 1
(arccos ρ) / π
½-½ρ
≈ .878
1
History
Intrinsic? Coincidence?
G = (V,E)
―> geometric problem
―> random cut
[Goemans-Williamson ’94]
GW-approximation,
GW = min
-1 < ρ < 1
(arccos ρ) / π
½-½ρ
≈ .878
History
[Bellare-Goldreich-Sudan ’92]
[Håstad ’97]
more than 83/84 is NP-hard
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
the conjectures
 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…
Unique Games 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!
Majority is Stablest conjecture:
How we want you to
interpret 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.
More motivation for result
o Provides insight to Unique Games conjecture.
o Fourier methods and related results independently
interesting.
o Motivates algorithmic progress on MAX-2SAT, MAX2LIN(q)
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:
o Hardness for MAX-CUT, from Unique Games
conjecture + long-code test
o Discussion of “Maj is Stablest” and partial results.
o Discussion of Unique Games conjecture.
The long-code
Encodes elements in {1,2,..,q}
The encoding of 2{1,2,3}:
1 1 1
1 1 -1
1 -1 1
1 -1 -1
1 -1 -1
…
1
1
-1
-1
1
..
In general, i{1,..,q} is encoded by f:{1,-1}q{-1,1},
defined by f(x)=xi
the GW algorithm
G=(V,E):
v
u
xv
maximize
1- xu , xv

2
(u,v)E
(*)
=

(u,v)E
xu  xv
4
2
(unit sphere in Rn)
xu
the GW algorithm
G=(V,E):
u
In S0, this is
Max-Cut!
v
xv
xu
maximize
(*)
1- xu , xv

2
(u,v)E
 opt
the GW algorithm
G=(V,E):
u
v
xv
maximize
(*)
1- xu , xv

2
(u,v)E
xu
GW algorithm: performance
Pr[(xu,xv) is cut]=
arccos(<xu,xv>)/
E[w(cut)] 

(u,v)E
(*)
arccos( xu , xv )/ 
1- xu , xv

2
(u,v)E
xv
xv n-1
xvS
<xu,xv>
xu
arccos(<xu,xv>)
donation to (*)
xu
GW algorithm: performance
Pr[(xu,xv) is cut]=
arccos(<xu,xv>)/
E[w(cut)] 

(u,v)E
(*)
arccos( xu , xv )/ 
1- xu , xv

2
(u,v)E
Actually this is
x
v
tight..
 arccos() /  
GW  min  arccos(<xu,xv>)
 (1  ) /2  xu
 0.879.. (for some  )
<xu,xv>
donation to (*)
Tight, if all inner
products are ρ*
Important example: Gρ
V = Sn-1
ρ - negative
E = {(x,y) : <x,y>  ρ}
[FS]
a hyperplane cut
is optimal for Gρ
size of cut:
(arccos ρ)/π
More important example: Dρ
V = {-1,1}n  Sn-1
a random edge (x,y):
y:
yi =
xi
-xi
well, actually {-n-½,n½}n
x~{-1,1}n,
w.p. ½ + ½ρ
w.p. ½ - ½ρ
higher probability
w(x,y) = P[(x,y) is chosen]
E[<x,y>] = ρ
tightly concentrated
Are Dρ and Gρ similar?
OPT(Dρ) = OPT(Gρ) = (arccos ρ)/π?
Dictatorship
no…
For f(x) = x7,
w( f-1(1),f-1(-1) ) = P[x7 ≠ y7]
=½-½ρ
For f(x) = sign(xi) = Maj(x),
w( f-1(1),f(-1) )=P[Maj(x)≠Maj(y)]
≈ (arccos ρ)/π
not at all a dictatorship
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)
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)
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:
.
soundness
completeness
=
(arccos ρ)/π
½ - ½ρ
≈ .878 (for ρ = ρ*)
Vol.2: Main results
Subhash Khot
IAS
Guy Kindler
DIMACS
Elchanan Mossel
UC Berkeley
Ryan O’Donnell
IAS
Unique Games Conjecture
“Unique Label Cover” with q colors:
Labels
[q]
n
πuv
Bijections
πuv
Input
πuv :
Unique Games Conjecture
“Unique Label Cover” with q colors:
Labels
[q]
n
πuv
Bijections
πuv
Assignment
πuv :
Unique Games Conjecture
Conjecture:
satisfying fraction of edges is hard, even if 1- of
them can be satisfied.
Labels
[q]
n
πuv
Bijections
πuv
Assignment
πuv :
Unique Games Conjecture
Conjecture:
satisfying fraction of edges is hard, even if 1- of
them can be satisfied.
o UGC is stronger than AS+ALMSS+Raz altogether.
UGC implies
o MIN-2SAT-Deletion hard to approximate to within any
constant factor [Hastad, Khot ’02]
o Vertex-Cover hard to approximate to within any factor
smaller than 2
[Khot-Regev ’03]
o These results need long-code tests, relying on theorems
in Fourier analysis.
[Bourgain ’02; Friedgut ’98]
Main theorem – proof overview
Assignment --> Cut
fv
fu
Max-Cut Test: Verify
fv(x)≠fu(y)
Main theorem – proof overview
Assignment --> Cut
fv
x
πuv (y)
Max-Cut
Test:
Verify
Max-Cut
Test:
Verify fv(x)≠f
fv(x)≠f
(y)(y))
u(πuuv
Completeness: at least (1-)(1-ρ)/2
Soundness: at most (1-`)(arccos ρ)/
fu
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.
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.