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On the Unique Games Conjecture. Subhash Khot NYU Courant CCC, June 10, 2010. Approximation Algorithms. A C-approximation algorithm for an NP-complete problem computes (C > 1), for problem instance I , solution A(I) s.t. Minimization problems :

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On the Unique Games Conjecture

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On the unique games conjecture

On the Unique Games Conjecture

Subhash Khot

NYU Courant

CCC, June 10, 2010


Approximation algorithms

Approximation Algorithms

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


Pcp theorem

PCP Theorem

[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.


In approximability towards tight hardness results

(In)approximability : Towards Tight Hardness Results

  • [Hastad’96]Clique n1-

  • [Hastad’97] MAX-3SAT 8/7 - 

  • [Feige’98] Set Cover (1- ) ln n

[Dinur’05]Combinatorial Proof of PCP Theorem !


Open problems in in approximability

Open Problems in (In) Approximability

  • 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]

    ………………………..


Unique games conjecture k 02

Unique Games Conjecture [K’02]

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)


Example of unique game 2csp

Example of Unique Game (2CSP)

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)

………….

………….


Unique game 2csp w permutation constraints

Unique Game 2CSP w/ Permutation Constraints

variable

k labels

Here k=4

constraints 


Unique game 2csp w permutation constraints1

Unique Game2CSP w/ Permutation Constraints

variable

k labels

Here k=4

Permutations or matchings

 : [k]  [k]


On the unique games conjecture

Unique Game

Find a labeling

that satisfies

max # constraints

OPT(G) = 6/7


Unique games

Unique Games

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.


Unique games conjecture

Unique Games Conjecture

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 ].


Supporting evidence

Supporting Evidence

[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”.


Small set expansion conjecture

Small Set Expansion Conjecture

[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.


On the 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


Linear equations over reals k moshkovitz 10

Linear Equations Over Reals [K Moshkovitz’10]

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


Unique games conjecture k 021

Unique Games Conjecture [K’02]

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)


Generic reduction from unique game bgs 95 long code hastad 97 fourier ugc

Generic Reduction from Unique Game[BGS’95 (Long Code), Hastad’97 (Fourier), UGC , ……]


Generic reduction from unique game bgs 95 long code hastad 97 fourier ugc1

PCP Reduction

Generic Reduction from Unique Game[BGS’95 (Long Code), Hastad’97 (Fourier), UGC , ……]

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) / 


Gadget dictatorships long codes

x

y

{-1,1} k

Gadget : Dictatorships (Long Codes)

  • 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.


Gadget cut that commits to co ordintae i

xi = 1

xi = -1

Gadget: Cutthat“commits” toco-ordintae i

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

= 

Observation : These are the maximum cuts.


Gadget cuts not committing to a co ordinate

Gadget : Cuts not committing to a co-ordinate

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) / .


Integrality gap

Integrality Gap

Given : Maximization Problem +

SDP relaxation.

  • For every problem instance G,

    SDP(G)  OPT(G)

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


Raghavendra 08

[Raghavendra’ 08]

  • 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.


Unique games conjecture k 022

Unique Games Conjecture [K’02]

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)


Inapproximability and fourier analysis

Inapproximability and Fourier Analysis

  • 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).


Inapproximability and fourier

Inapproximability and Fourier

  • 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.


Inapproximability and fourier1

Inapproximability and Fourier

  • 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 ?


Unique games conjecture k 023

Unique Games Conjecture [K’02]

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)


Disproving ugc means

Disproving UGC means ..

For small enough (constant),

given a UG with optimum 1- ,

algorithm that finds a labeling satisfying

(say) 50% constraints,

irrespective of k = #labels.


Algorithmic results

Algorithmic Results

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 …


If the ugc is true then

If the UGC is true, then :

  • 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.


Sdp relaxation of unique games fl 92

Orthonormal

Bases for Rk

v1 , v2 ,

… , vk

v

variables

k labels

u

u1 , u2 ,

… , uk

Matchings [k]  [k]

SDP Relaxation of Unique Games [FL’92]

  • OPT(G) = 1- εSDP(G) ≥ 1- ε .

  • For i = 1, …, k,  ui , vi  ≥ 1- ε ,

    up to permutation of indices.


K 02 cmm 05 rounding algorithm

vk

v2

v1

uk

u2

u1

[K’02, CMM’05] Rounding Algorithm

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.


Trevisan 05 algorithm

[Trevisan’05] Algorithm

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.


Akkstv 08 algorithm

[AKKSTV’08 Algorithm]

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.


On the unique games conjecture

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


Arora barak steurer 10 algorithm kolla 10 naor 10

[Arora Barak Steurer’10 Algorithm][Kolla’10, Naor’10]

  • 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.


Arora barak steurer 10 algorithm kolla 10 naor 101

[Arora Barak Steurer’10 Algorithm][Kolla’10, Naor’10]

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) ).


Unique games conjecture k 024

Unique Games Conjecture [K’02]

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)


Gaussian isoperimetry

(Gaussian) Isoperimetry

. [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 ).


Gaussian isoperimetry1

(Gaussian) Isoperimetry

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.


Integrality gap1

Integrality Gap

  • [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.


Kv 05 integrality gap for unique games sdp

Orthonormal

Bases for Rk

v1 , v2 ,

… , vk

v

variables

k labels

u

u1 , u2 ,

… , uk

Matchings [k]  [k]

[KV’05] Integrality Gap for Unique Games SDP

SDP(G) = 1-o(1)

Unique Game G with

OPT(G) = o(1)


Integrality gap for max cut with triangle inequality

u1 , u2 ,

… , uk

Integrality Gap for MAX-CUT with Triangle Inequality

OPT(G) = o(1)

PCP Reduction

No large cut

Good MAX-CUT SDP solution

 u1  u2  u3 ……… uk-1  uk

{-1,1}k


Max cut and sparsest cut i g

MAX-CUT and Sparsest Cut I.G.

  • [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.


Open problems

Open Problems

  • 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?


Unique games conjecture k 025

Unique Games Conjecture [K’02]

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)


Gap amplification

Gap Amplification

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.


Strong parallel repetition

Strong Parallel Repetition ?

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.


Raz s example optimal foam

Raz’s Example OptimalFoam

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


Conclusion

Conclusion

  • (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 …


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