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On the Unique Games Conjecture

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

- Max Cut
(17/16 vs 1/0.878… )

[Håstad’97, GoemansWilliamson’94]

………………………..

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%

variables

constraints

variables

k labels

Here k=4

constraints

variables

k labels

Here k=4

Constraints = Bipartite graphs

or Relations [k] [k]

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, ….

What about unique games ?

variable

k labels

Here k=4

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)
- [Hastad’97]
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])

Gadget constructed via Fourier theorem

+

Connecting gadgets via Unique Game instance

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

entire construction to the analysis

of the gadget”.

Gadget =

Basic gadget --->

Bipartite gadget --->

Bipartite gadget with permutation

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

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

x = 011

k = # labels

{0,1} k

Y = 110

{0,1} k

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

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

gadget)

x = 011

y’ = 110

{0,1} k

{0,1} k

Matching co-ordinate cuts have size =

x = 011

Y ’ = 110

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]

{0,1} k

Vertices

Edges

Bipartite

Gadget

via

(Completeness) :

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

Unique Game Graph H

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)

x

{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)

{0,1} k

Vertices

Edges

Bipartite

Gadget

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.

Basic Gadget ----> Bipartite Gadget with

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.

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

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.

r

vk

v2

r

v1

uk

u2

u1

- 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

- The UGC
- Hardness of Approximation Results
- I hope UGC is true
- Attempts to Disprove : Algorithms
Connections/applications :

- Fourier Analysis
- Integrality Gaps
- Metric Embeddings

- 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 !

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 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

[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

(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

Bin-packing (additive approximation)

……………………

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]