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On the Unique Games Conjecture

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On the Unique Games Conjecture

Subhash Khot

NYU Courant

CCC, June 10, 2010

A C-approximation algorithm for an NP-complete

problem computes (C > 1),

for problem instance I , solution A(I) s.t.

Minimization problems :

A(I) C OPT(I)

Maximization problems :

A(I) OPT(I) / C

[B’85, GMR’89, BFL’91, LFKN’92, S’92,……] [PY’91]

[FGLSS’91, AS’92 ALMSS’92]

Theorem : It is NP-hard to tell whether a

MAX-3SAT instance is

* Satisfiable (i.e. OPT = 1) or

* No assignment satisfies more than

99% clauses

(i.e. OPT 0.99).

i.e. MAX-3SAT is 1.01 hard to approximate.

- [Hastad’96]Clique n1-
- [Hastad’97] MAX-3SAT 8/7 -
- [Feige’98] Set Cover (1- ) ln n

[Dinur’05]Combinatorial Proof of PCP Theorem !

- Vertex Cover
(1.36 vs. 2) [DinurSafra’02]

- Coloring 3-colorable graphs
(5 vs. n3/14)

[KhannaLinialSafra’93, BlumKarger’97]

- Sparsest Cut
(1+ε vs. (logn)1/2) [AMS’07,AroraRaoVazirani’04]

- Max Cut
(17/16 vs 1/0.878… )

[Håstad’97, GoemansWilliamson’94]

………………………..

Supporting Evidence

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

OPT = max fraction of equations that can

be satisfied by any assignment.

x1 - x3 = 2 (mod k)

x5 -x2 = -1 (mod k)

x2 - x1 = k-7 (mod k)

………….

………….

variable

k labels

Here k=4

constraints

variable

k labels

Here k=4

Permutations or matchings

: [k] [k]

Unique Game

Find a labeling

that satisfies

max # constraints

OPT(G) = 6/7

Considered before ……

[Feige Lovasz’92] Parallel Repetition of UG

reduces OPT(G).

How hard is approximating OPT(G) ?

Observation : Easy to decide whether

OPT(G) = 1.

For any , , there is integer k(, ), s.t.

it is NP-hard to tell whether a Unique

Game with k = k(, ) labels has

OPT 1-

or OPT

i.e. Gap-Unique Game (1- , ) is NP-hard.

Gap Projection Game (1, ) is NP-hard.

[ PCP Theorem + Raz’s Parallel Repetition Theorem ].

[UGC] Gap-Unique Game (1-ε, ) is NP-hard.

[Feige Reichman’04] Gap-Unique Game (C, ) is NP-hard.

However C --> 0 as C --> ∞.

[KV’05] SDP relaxation for UG has “integrality gap” (1-, ).

[KV’05] UGC based predictions were proven correct.

Specifically, metric embedding lower bounds.

[Wishful thinking] “There is structure in CS/math”.

[Raghavendra Steurer’ 10]Φ (S ) = Edge expansion of set S.

For every ε > 0, there exists δ > 0, such that,

it is NP-hard to tell whether in a graph G(V,E),

- There is a set S, |S| = δ |V|,Φ (S) ≤ ε.
- For every set S, |S| ≈ δ |V|,Φ (S) ≥ 1- ε.
[Raghavendra Steurer’ 10]

SSE Conjecture Unique Games Conjecture.

Unique Game and Small Set Expansion

|G’| = n k.

S = Optimal labeling.

|S| = 1/k |G ’|.

Φ(S) = 1- OPT(G).

Unique Game G with

n variables, k labels

Homogeneous 3LIN(R): x1 – x3 + 2 x5 = 0.

∙∙∙∙∙∙∙∙

eq: xi + .5 x j -x k = 0.

Theorem: It is NP-hard to tell if :

- There is a “non-trivial” solution that satisfies
1-ε fraction of equations.

- Any “non-trivial” solution fails on a constant
fraction of equations with error Ω(√ε).

3LIN(R) to 2LIN(R) reduction ? 2LIN(R) ≈ Sparsest Cut

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

PCP Reduction

MAX-CUT Instance

Unique Game Instance

k labels

Gadget: {-1,1} k

Match Goemans-Williamson’s

SDP rounding Algorithm

1/0.878… Hardness

OPT(UG) > 1-ε sized cut.

OPT(UG) < δ No cut with size

arccos (1-2) /

x

y

{-1,1} k

- Consider f: {-1, 1}k {-1,1}, i.e. Cuts.
- Encode label i Є {1,2,…., k} by dictatorship function
f(x) = xi.

Weighted graph, total edge weight = 1.

Picking random edge :

x R{-1,1} k

y <-- flip every co-ordinate of x with

probability ( 0.8)

Noise-sensitivity graph.

xi = 1

xi = -1

Fraction of edges cut = Pr(x,y) [xi yi ]

=

Observation : These are the maximum cuts.

Influence (i, f) = Prx [ f(x) f(x+ei) ]

How large can be cuts with no influential

co-ordinate ?

Random Cut : ½

Majority Cut : > arccos (1-2) / > ½

[KKMO’04, MOO’05]Majority Is Stablest (Under Noise)

Any cut with no influential co-ordinate has size

at most arccos (1-2) / .

Given : Maximization Problem +

SDP relaxation.

- For every problem instance G,
SDP(G) OPT(G)

- Integrality Gap = Sup G SDP(G) / OPT(G)

- Duality between Algorithms and Hardness.
- For every CSP, write a natural SDP relaxation.
- Integrality gap = β. Implies β-approximation.
- Theorem: Every instance with gap β’ < β can
be used to construct a gadget and prove

UGC-based β’- hardness result !

- SDPs are optimal algorithm for CSPs.

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

- f : {-1,+1} k {-1,+1}, balanced.
- Sparsest Cut[KV’05, CKKRS’05]
[KKL’88] f has a co-ordinate with influence Ω(log k /k).

[Bourgain’02] If NSε(f) << √ε, then f depends

essentially on exp(1/ε2) co-ordinates.

- MAX-CUT[KKMO’04]Majority Is Stablest
[MOO’05] If f has no influential co-ordinate, then

NS ε(f) ≥ NS ε(Majority) - o(1).

- f : {-1,+1}k {-1,+1}, balanced.
- Vertex Cover[DinurSafra’02, K Regev’03, K Bansal’09]
[Friedgut’98] If total influence is k, then f depends

essentially on exp(k) co-ordintaes.

[MOO’05]It Ain’t Over Till It’s Over

If f has no influential co-ordinate, then

on almost every subcube of {-1, +1}k of dimension k/100,

f = 1 and f = -1 with constant probability.

- f : {-1,+1}k {-1,+1}, balanced.
- MAX-k-CSP [Samorodintsky Trevisan ’06]
If f has no influential co-ordinate, then f has low

Gowers’ Uniformity norm.

Open: f: [q] k [q], q ≥ 3, no influential co-ordinate.

- f balanced. Is Plurality Stablest ?
- What is the maximum Fourier mass at the first level ?

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

For small enough (constant),

given a UG with optimum 1- ,

algorithm that finds a labeling satisfying

(say) 50% constraints,

irrespective of k = #labels.

Algorithm that finds a labeling

satisfying f(, k, n) fraction of constraints.

[K’02] 1- 1/5 k2

[Trevisan’05]1- 1/3 log1/3 n

[Gupta Talwar’05] 1- log n

[CMM’05] 1/k , 1- 1/2 log1/2 k

[CMM’06] 1- log1/2 k log1/2 n

[AKKSTV’08 , Kolla’10] UG on “mild” expander graphs.

[ABS’10] Exp ( n ) time algorithm.

None of these disproves UGC. However …

- k >> 21/ε .
- Graph of constraints cannot even be a “mild”
expander. UG is easy on random graphs.

- Reduction from 3SAT must blow up the size
by n1/ε .

- Conjecture does not hold for sub-constant ε,
i.e. below 1/log n.

Orthonormal

Bases for Rk

v1 , v2 ,

… , vk

v

variables

k labels

u

u1 , u2 ,

… , uk

Matchings [k] [k]

- OPT(G) = 1- εSDP(G) ≥ 1- ε .
- For i = 1, …, k, ui , vi ≥ 1- ε ,
up to permutation of indices.

vk

v2

v1

uk

u2

u1

r

r

Random r

u

v

Pick the label closest to r. Label(u) = Label(v) = 2.

Pr [ Label(u) = Label(v) ] > 1 - 1/5 k2[K’02].

Pr [ Label(u) = Label(v) ] > 1- 1/2 log1/2 k[CMM’05].

- Labeling satisfies 1- 1/2 log1/2 k fraction
of constraints in expected sense.

Graph of variables and

constraints

- [Leighton Rao’88] Delete 1% of edges so that
all connected components have diameter O(log n).

- Algorithm to solve UG on low diameter graph.

Algorithm that works on a UG instance s.t.

- 1-ε satisfiable and,
- Every balanced cut in the graph cuts at least
Ω ( √ε ) fraction of edges.

- SDP-based.
- “Mild” expansion Almost all SDP vector
tuples are nearly identical

Yields a good labeling.

Unique Game and Small Set Expansion

Label extended Graph

|G’| = n k.

S = Optimal labeling.

|S| = 1/k |G ’|.

Φ(S) = 1- OPT(G).

UG G with

n variables, k labels

- Algo. runs in time exp(nε) on UG that is 1-ε satisfiable.
- Good solution to UG Small non-expanding set S in G’.
- Small non-expanding set in label-extended graph G’
Either corresponds to a good UG solution (useful)

Or is a non-expanding set in G (fake).

- Iteratively remove all fake sets from G, sacrificing at
most 1% edges.

Main Lemma (Algorithmic) :

If every set of size n1-ε expands by Ω(ε2), then the

number of eigenvalues exceeding 1-ε is nO(ε).

- The UG solution is found in the linear span of
eigenvectors with eigenvalues ≥ 1-ε. [Kolla’10]

- Run-time exp ( nO(ε) ).

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

. [MOO’05]Majority Is Stablest reduces via Invariance Principle,

to a geometric question:

P: Rn {-1,+1} be a partition of Gaussian space

into two sets of equal measure.

NSε(P) = Pr [ P(x) ≠ P(y) ], Cor (x,y) = 1-2ε.

Which P minimizes the noise-sensitivity?

[Borell’85] NSε(P) ≥ NSε( HALF-SPACE THRU ORIGIN ).

Open: q ≥ 3. More Invariance.

[IM’10] MAX-q-CUT Problem.

Plurality is Stablest Conjecture.

Partition Rn into q equal parts.

(Geometric): Standard Simplex Conjecture.

[K Naor’08]Kernel ClusteringProblem.

Maximizing Fourier Mass at First Level.

(Geometric): Propeller Conjecture.

- [Feige Schechtman’01] [Goemans Williamson’92]
1/0.878.. Integrality gap for MAX-CUT.

- SDP with “triangle inequality constraints” ?
- ω(1) Integrality gap for Sparsest Cut?
- UGC NP-hardness
These integrality gaps exist.

Orthonormal

Bases for Rk

v1 , v2 ,

… , vk

v

variables

k labels

u

u1 , u2 ,

… , uk

Matchings [k] [k]

SDP(G) = 1-o(1)

Unique Game G with

OPT(G) = o(1)

u1 , u2 ,

… , uk

OPT(G) = o(1)

PCP Reduction

No large cut

Good MAX-CUT SDP solution

u1 u2 u3 ……… uk-1 uk

{-1,1}k

- [KV’05] MAX-CUT gap matching Goemans-Williamson
even with triangle inequality constraints.

- [KV’05, KrauthgamerRabani’05, DKSV’06]
(loglog n) integrality gap for Sparsest Cut SDP.

An n-point “negative type” metric that needs

distortion (loglog n) to embed into L1.

Refutation of [Goemans Linial’97, ARV’04] conjectures.

- [KS’09, RS’09] Similar gaps for SDP + Sherali-Adams LP.
Negative type metric that is L1 embeddable

locally but not globally.

- Integrality gaps for the Lasserre SDP Relaxation?
Lasserre Relaxation could potentially disprove UGC.

- Sparsest Cut (NEG versus L1 Metrics) :
[ARV’04, AroraLeeNaor’05] O(√log n).

[LeeNaor’06, CheegerKleiner’06, CKNaor’09].

Ω(logc n), c = ½?

Connections to:

- Inapproximability (UGC several problems inapproximable)
- Discrete Fourier Analysis (e.g. KKL, Majority is Stablest)
- Geometry (Isoperimetry, Metric geometry, Integrality gaps)
- Algorithms (Attempts to disprove UGC)
- Parallel Repetition (Gap amplification, Foam construction)

Prove UGC in two steps (?):

- Prove “mild” hardness, i.e.
GapUG (1-ε’ , 1-ε’’ ) is hard.

- Amplify gap via parallel repetition to
GapUG (1-ε , δ).

Note however that even proving “mild” hardness

is a huge challenge.

OPT(G) = 1-ε.

[Raz’98] OPT(Gm) ≤ (1-ε32 )m/log k . 2P1R Games

[Holenstein’07] OPT(Gm) ≤ (1-ε3 )m/log k . 2P1R Games

[Rao’08] OPT(Gm) ≤ (1-ε2)m . Projection Games (UG).

GapUG (1-ε , δ ) is NP-hard iff

GapUG (1-ε , 1 - √ε C(ε) ) is NP-hard where

C(ε) –> ∞ as ε –> 0.

[Raz’08] The rate (1-ε2)m cannot be improved further.

Problem: Tiling Rd using a “shape” of unit

volume and minimum surface area.

[Kindler O’Donnell Rao Wigderson ’08, Alon Klartag’09]

There exists a tiling shape with unit

volume and surface area O(√d ) !

Great for tiling:

Surface area = 2d

Not good for tiling:

Surface area ≈ √d

- (Dis)Prove Unique Games Conjecture.
- Intermediate between P and NP-complete?
- Prove hardness results bypassing UGC.
- TSP, Steiner Tree, Scheduling Problems ?
- More techniques, connections, results …