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On the Unique Games Conjecture. Subhash Khot Georgia Inst. Of Technology. At FOCS 2005. NP-hard Problems. Vertex Cover MAX-3SAT Bin-Packing Set Cover Clique MAX-CUT …………….. ……………. Approximability : Algorithms .

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### On the Unique Games Conjecture

Subhash Khot

Georgia Inst. Of Technology.

At FOCS 2005

• Vertex Cover

• MAX-3SAT

• Bin-Packing

• Set Cover

• Clique

• MAX-CUT

• ……………..

• ……………..

A C-approximation algorithm 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

• Vertex Cover2 - approx.

• MAX-3SAT8/7 - approx. Random assignment.

• Packing/Scheduling(1+) – approx.   > 0 (PTAS)

• Set Coverln n approx.

• Clique n/log n [Boppana Halldorsson’92]

• Many more , ref. [Vazirani’01]

[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 1/0.99 = 1.01 hard to approximate.

i.e. MAX-3SAT and MAX-SNP-complete problems [PY’91]

have no PTAS.

• [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 vs. (logn)1/2) [AroraRaoVazirani’04]

• Max Cut

(17/16 vs 1/0.878… )

………………………..

Unique Games Conjecture [Khot’02]

Implies these hardness results :

• Vertex Cover 2-  [KR’03]

• Coloring 3-colorable (1) [DMR’05]

graphs (variant of UGC)

• MAX-CUT 1/0.878.. -  [KKMO’04]

• Sparsest Cut,

Multi-cut [KV’05,

(1) CKKRS’04]

Min-2SAT-Deletion [K’02, CKKRS’04]

Led to …

[MOO’05] Majority Is Stablest Theorem

[KV’05] “Negative type” metrics do not embed

into L1 with O(1) “distortion”.

Optimal “integrality gap” for MAX-CUT

SDP with “Triangle Inequality”.

Given : Maximization Problem +

Specific SDP relaxation.

• For every problem instance G,

SDP(G)  OPT(G)

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

• Constructing gap instance = negative result.

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

• A maximization problem called “Unique Game” is hard to approximate.

• “Gap-preserving” reductions from

Unique Game 

Hardness results for Vertex Cover, MAX-CUT, Graph-Coloring, …..

OPT = max fraction of equations that can

be satisfied by any assignment.

x1 + x3 = 2 (mod k)

3 x5 -x2 = -1 (mod k)

x2 + 5x1 = 0 (mod k)

UGC For large k, it is NP-hard to tell

whether OPT 99% or

OPT  1%

2-Prover-1-Round Game (Constraint Satisfaction Problem )

variables

constraints 

2-Prover-1-Round Game (Constraint Satisfaction Problem )

variables

k labels

Here k=4

constraints 

2-Prover-1-Round Game (Constraint Satisfaction Problem )

variables

k labels

Here k=4

Constraints = Bipartite graphs

or Relations   [k]  [k]

2-Prover-1-Round Game (Constraint Satisfaction Problem )

Find a labeling

that satisfies

max # constraints

variables

k labels

Here k=4

OPT(G) = 7/7

• Given a 2P1R game G, how hard

is it to find OPT(G) ?

• PCP Theorem + Raz’s Parallel Repetition Theorem :

For every , there is integer k(), s.t.

it is NP-hard to tell whether a 2P1R

game with k = k() labels has

OPT = 1 or OPT  

In fact k = 1/poly()

• Almost all known hardness results

(e.g. Clique, MAX-3SAT, Set Cover, SVP, …. )

are reductions from 2P1R games.

• Many special cases of 2P1R games are known to be hard, e.g. Multipartite graphs,

Expander graphs,

Smoothness property, ….

Unique Game = 2P1R Game with Permutations

variable

k labels

Here k=4

Unique Game = 2P1R Game with Permutations

variable

k labels

Here k=4

Permutations or matchings

 : [k]  [k]

Unique Game = 2P1R Game with Permutations

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)

for a unique game G ?

Observation : Easy to decide whether

OPT(G) = 1.

• Vertices : Binary variables x, y, z, w, …….

• Edges : Equations x + y = 1 (mod 2)

NP-hard to tell whether

OPT(MAX-CUT)  17/21

or OPT(MAX-CUT)  16/21

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.

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

• Given a graph, find a cut that maximizes

fraction of edges cut.

• Random cut : 2-approximation.

• [GW’94] SDP-relaxation and rounding.

min 0 <  < 1  / (arccos (1-2) /  )

= 1/0.878 … approximation.

• [KKMO’04] Assuming UGC, MAX-CUT is

1/0.878… -  hard to approximate.

Unique Game Graph H

• Completeness :

OPT(UG) > 1-o(1)    - o(1) cut.

• Soundness :

OPT(UG) < o(1)  No cut with

size arccos (1-2) /  + o(1)

• Hardness factor =  / (arccos (1-2) /  ) - o(1)

• Choose best  to get 1/0.878 … (= [GW’94])

+

Connecting gadgets via Unique Game instance

[DMR’05]“UGC reduces the analysis of the

entire construction to the analysis

A graph on {0,1} k with specific properties

(e.g. cuts, vertex covers, colorability)

x = 011

k = # labels

{0,1} k

Y = 110

y

Weighted graph, total edge weight = 1.

Picking random edge :

x R{0,1} k

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

probability  (  0.8)

x

xi = 0

xi = 1

MAX-CUT Gadget : Co-ordinate CutAlong Dimension i

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

= 

Observation : These are the maximum cuts.

A graph on {0,1} k  {0,1} k (double cover of basic

x = 011

y’ = 110

{0,1} k

{0,1} k

Matching co-ordinate cuts have size = 

Y ’ = 110

Bipartite Gadget with Permutation  : [k] -> [k]

Co-ordinates in second hypercube permuted via .

Example :  = reversal of co-ordinates.

 (y’) = 011

OPT  1 – o(1)

or OPT o(1)

Variables

k labels

Permutations  : [k]  [k]

Reduction from Unique Game

Vertices

Edges

Instance H of MAX-CUT

Bipartite

via 

(Completeness) :

OPT(UG) > 1-o(1)  H has  - o(1) cut.

Unique Game Graph H

Completeness : OPT(UG)  1-o(1)

label = 1

Labels = [1,2,3]

label = 2

label = 3

label = 2

label = 1

label = 1

label = 3

Completeness : OPT(UG)  1-o(1)

{0,1} k

Vertices

Edges

Hypercubes are cut along dimensions = labels.

MAX-CUT   - o(1)

Unique Game Graph H

(Soundness) :

OPT(UG) < o(1)  H has no cut of

size arccos (1-2) /  + o(1)

{0,1} k

y

Cuts = Boolean functions f : {0,1} k  {0,1}

Compare boolean functions

* that depend only on single co-ordinate vs

* where every co-ordinate has negligible

“influence” (i.e. “non-junta” functions)

f(x1 x2 …….. xk) = xi

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

f(x1 x2 …….. xk) = MAJORITY

How large can non-junta cuts be ?

i.e. cuts with all influences negligible ?

Random Cut : ½

Majority Cut : arccos (1-2) /  > ½

• [MOO’05]Majority Is Stablest (Best)

Any cut slightly better than

Majority Cut must have

“influential” co-ordinate.

{0,1} k

{0,1} k

[MOO’05] Any “special” cut with value

arccos (1-2) /  +  must define a

matching pair of influential co-ordinates.

{0,1} k

{0,1} k

f : {0,1} k --> {0, 1}

g : {0,1} k --> {0, 1}

cut > arccos (1-2) /  + 

 i Infl (i, f), Infl (i, g) > (1)

Vertices

Edges

Instance H of MAX-CUT

Bipartite

via 

• Assume arccos (1-2) /  +  cut exists.

• On /2 fraction of constraints, the

bipartite gadget has arccos (1-2) /  + /2 cut.

 matching pair of labels on this constraint.

This is impossible since OPT(UG) = o(1).

Done !

• Vertex Cover

Friedgut’s Theorem

Every boolean function with low “average sensitivity” is a junta.

• Sparsest Cut, Min-2SAT Deletion KahnKalaiLinial Every balanced boolean function has a

co-ordinate with influence log n/n.

Bourgain’s Theorem(inspired by Hastad-Sudan’s 2-bit Long Code test)

Every boolean function with low “noise sensitivity” is a junta.

• Coloring 3-Colorable [MOO’05] inspired.

Graphs

•  Hardness results for Clique, MAX-3SAT, …….

• Instead of Unique Games, use reduction from

general 2P1R Games (PCP Theorem + Raz).

• Hypercube = Bits in the Long Code [Bellare

Goldreich Sudan’95]

• PCPs with 3 or more queries (testing Long Code).

• Not enough to construct 2-query PCPs.

Power in simplicity.

“Obvious” way of encoding a permutation

constraint.

permutation.

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

• Implies all the “right” hardness

results in a unifying way.

• Neat applications of Fourier theorems

[Bourgain’02, KKL’88, Friedgut’98, MOO’05]

• Surprising application to theory of metric

embeddings and SDP-relaxations [KV’05].

• Mere coincidence ?

[Feige Reichman’04]

Gap-Unique Game (C, ) is NP-hard.

i.e. For every constant C, there is  s.t.

it is NP-hard to tell if a UG has

OPT > C  or OPT < .

However C  --> 0 as  --> 0.

[Khot Vishnoi’05]

SDP relaxation for Unique Game

has integrality gap (1-, ).

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

For small enough (constant),

given a UG with optimum 1- ,

algorithm that finds a labeling satisfying

(say) 50% constraints.

Algorithm that finds a labeling

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

[Khot’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

None of these disproves UGC.

Quadratic Integer Program For Unique Game [Feige Lovasz’92]

variable

u1 , u2 , … , uk  {0,1}

u

 : [k]  [k]

v

k labels

v1 , v2 , … , vk  {0,1}

vi = 1 if Label(v) = i

= 0 otherwise

Constraints on edge-set E.

• Maximize   ui vπ(i)

(u, v)  E i=1,2,..,k

•  u  i  [k], ui  {0,1}

•  u  ui2 = 1

i

• u  i ≠ j , ui uj = 0

• Maximize   ui, vπ(i) 

(u, v)  E i=1,2,..,k

•  u  i  [k], ui is a vector.

•  u  || ui ||2 = 1

i=1,2,..,k

•  u  i≠j  [k], ui, uj = 0

• OPT(G)  SDP(G)  1.

• If OPT(G) < 1, then SDP(G) < 1.

• SDP(Gm) = (SDP(G))m

• Parallel Repetition Theorem for UG :

OPT(G) < 1  OPT(Gm)  0

vk

v2

v1

uk

u2

u1

[Khot’02] Rounding Algorithm

r

r

Random r

u

v

• Label(u) = 2, Label(v) = 2

• Pr [ Label(u) = Label(v) ] > 1 - 1/5 k2

• Labeling satisfies 1 - 1/5 k2 fraction

of constraints in expected sense.

vk

v2

r

v1

uk

u2

u1

[CMM’05] Algorithm

• Labeling that satisfies 1/k fraction

of constraints. (Optimal [KV’05])

All i s.t. ui is “close” to r are taken

as candidate labels to u.

Pick one of them at random.

• Given a unique game with optimum

1- 1/log n, algorithm finds a labeling

that satisfies 50% of constraints.

• Limit on hardness factors achievable

via UGC (e.g. loglog n for Sparsest Cut).

Variables and constraints

[Leighton Rao’88] Delete a few constraints and

remaining graph has connected

components of low diameter.

A good algorithm for graphs with low

diameter.

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

Already Covered Let’s move on ….

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

[KV’05] Integrality Gaps for SDP-relaxations

• MAX-CUT

• Sparsest Cut

• Unique Game

Gaps hold for SDPs with “Triangle Inequality”.

Given G(V,E)

• Maximize ¼  |vi - vj |2

(i, j)  E

•  i, vi  {-1,1}

• Triangle Inequality (Optional) :  i, j , k,

|vi - vj |2 + |vj - vk |2 |vi - v k|2

• Maximize ¼  || vi - vj ||2

(i, j)  E

•  i, vi  Rn, || vi || = 1

• Triangle Inequality (Optional) :  i, j , k,

|| vi - vj ||2 + || vj - vk ||2 || vi - v k||2

• [Goemans Williamson’94]

Integrality gap  1/0.878..

• [Karloff’99] [Feige Schetchman ’01]

Integrality gap  1/0.878.. - 

SDP solution does not satisfy Triangle Inequality.

Does Triangle Inequality make the SDP tighter ?

NO if Unique Games Conj. is true !

Bases for Rk

v1 , v2 ,

… , vk

v

variables

k labels

u

u1 , u2 ,

… , uk

Matchings [k]  [k]

Integrality Gap for Unique Games SDP

SDP(G) = 1-o(1)

Unique Game G with

OPT(G) = o(1)

u1 , u2 ,

… , uk

Integrality Gap for MAX-CUT with Triangle Inequality

OPT(G) = o(1)

PCP Reduction

No large cut

Good SDP solution

 u1  u2  u3 ……… uk-1  uk

{-1,1}k

• The UGC

• Hardness of Approximation Results

• I hope UGC is true

• Attempts to Disprove : Algorithms

Connections/applications :

• Fourier Analysis

• Integrality Gaps

• Metric Embeddings

• Metric is a distance function on [n] such that

d(i, j) + d(j, k)  d(i, k).

• Metric d embeds into metric  with

distortion   1 if

 i, j d(i, j)   (i, j)   d(i, j).

Given a set of vectors satisfying Triangle Inequality :

 i, j , k,

|| vi - vj ||2 + || vj - vk ||2 || vi - v k||2

d(i, j) = || vi - vj ||2 defines a metric.

These are called “negative type metrics”.

L1  NEG  METRICS

NEG vs L1 Question

[Goemans, Linial’ 95]

Conjecture :NEG metrics embed into L1

with O(1) distortion.

O(1) Integrality Gap

O(1) Approximation

[Linial London

Rabinovich’94]

[Aumann Rabani’98]

[Chawla Krauthgamer Kumar

Rabani Sivakumar ’05]

[KV’05]

(1) hardness result

Unique Games

Conjecture

Sparsest Cut

NEGvs L1 Lower Bound

(loglog n) integrality gap for Sparsest

Cut SDP. [KhotVishnoi’05, KrauthgamerRabani’05]

 A negative type metric that needs

distortion (loglog n) to embed into L1.

• (Dis)Prove Unique Games Conjecture.

• Prove hardness results bypassing UGC.

• NEG vs L1 , Close the gap.

(log log n) vs (log n loglog n)

[Arora Lee Naor’04]

• Prove hardness of Min-Deletion version

of Unique Games. (log n approx. [GT’05])

• Integrality gaps with “k-gonal” inequalities.

• Is hypercube (Long Code) necessary ?

More hardness results, integrality gaps,

embedding lower bounds, Fourier Analysis, ……

[Samorodnitsky Trevisan’05] “Gowers Uniformity,

Influence of Variables, and PCPs”.

UGC  Boolean k-CSP is hard to

approximate within 2k- log k

Independent Set on degree D

graphs is hard to approximate

within D/poly(log D).

Traveling Salesperson

Steiner Tree

Max Acyclic Subgraph, Feedback Arc Set

……………………

Recent progress on Edge Disjoint Paths

Network Congestion

Shortest Vector Problem

Asymmetric k-center (log* n)

Group Steiner Tree (log2 n)

Hypergraph Vertex Cover

………………

System of linear equations mod k.

x1 + x3 = 2

3 x5 -x2 = -1

x2 + 5x1 = 0

[KKMO’04] UGC  UGC in the special case of

linear equations mod k.

• 2-to-1 Conjecture [K’02]

• -Conjecture [DMR’05]

 NP-hard to color 3-colorable graphs

with O(1) colors.

  [k]  [k]

  [k]  [k]