Yuan - PowerPoint Presentation

Slide1

Yuan Zhou, Ryan O’DonnellCarnegie Mellon University

Approximability

and

Proof ComplexitySlide2

Constraint Satisfaction ProblemsGiven:a set of variables: Va set of values: Ωa set of "local constraints": EGoal: find an assignment σ : V -> Ω to maximize #satisfied constraints in E

α

-approximation algorithm:

always outputs a solution of value

at least

α*OPTSlide3

Example 1: Max-CutVertex set: V = {1, 2, 3, ..., n}Value set: Ω = {0, 1}

Typical local constraint:

(

i

, j)

in

E

wants

σ

(

i

) ≠

σ

(j)

Alternative description:

Given G = (V, E), divide V into two parts,

to maximize #edges across the cut

Best approx. alg.:

0.878-approx.

[GW'95]

Best NP-hardness:

0.941

[Has'01, TSSW'00]Slide4

Example 2: Balanced SeparatorVertex set: V = {1, 2, 3, ..., n}Value set: Ω = {0, 1}

Minimize #satisfied local constraints:

(

i

, j)

in

E

:

σ

(

i

) ≠

σ

(j)

Global constraint: n/3 ≤ |{

i

:

σ

(

i

) = 0}| ≤ 2n/3

Alternative description:

given G = (V, E)

divide V into two "balanced" parts,

to minimize #edges across the cutSlide5

Example 2: Balanced Saperator (cont'd)Vertex set: V = {1, 2, 3, ..., n}Value set: Ω

= {0, 1}

Minimize #satisfied local constraints:

(

i

, j)

in

E

:

σ

(

i

) ≠

σ

(j)

Global constraint: n/3 ≤ |{

i

:

σ

(

i

) = 0}| ≤ 2n/3

Best approx. alg.:

sqrt

{log n}-approx.

[ARV'04]

Only

(1+

ε)-approx. alg.

is ruled out

assuming

3-SAT does not have

subexp

time alg.

[AMS'07]Slide6

Example 3: Unique GamesVertex set: V = {1, 2, 3, ..., n}Value set: Ω = {0, 1, 2, ..., q - 1}

Maximize #satisfied local constraints:

{

(

i

, j

), c}

in

E

:

σ

(

i

) -

σ

(j) = c (mod q)

Unique Games Conjecture (UGC)

[Kho'02, KKMO'07]

No poly-time algorithm, given an instance where optimal solution satisfies (1-ε) constraints, finds a solution satisfying

ε

constraints

Stronger than (implies) "no constant approx. alg."Slide7

Open questionsIs UGC true?Is Max-Cut hard to approximate better than 0.878?

Is

Balanced

Separator

hard to approximate with in constant factor

?

Easier questions

Do the known powerful optimization algorithms solve

UG

/

Max-Cut

/

Balanced Separator

?Slide8

SDP Relaxation hierarchiesA systematic way to write tighter and tighter SDP relaxationsExamplesSherali-Adams+SDP [SA'90]

Lasserre hierarchy

[Par'00, Las'01]

…

?

UG(ε)

-round

SDP relaxation in roughly time

BASIC-SDP

GW SDP for Maxcut (0.878-approx.)

ARV SDP for Balanced

SeparatorSlide9

How many rounds of tighening suffice?Upperbounds rounds of SA+SDP suffice for UG

[ABS'10, BRS'11]

Lowerbounds

[KV'05, DKSV'06, RS'09, BGHMRS '12]

(also known as constructing

integrality gap instances

)

rounds of SA+SDP needed for

UG

rounds of SA+SDP needed for better-than-0.878

approx

for

Max-Cut rounds for SA+SDP needed for constant approx. for Balanced SeparatorSlide10

From SA+SDP to Lasserre SDPAre the integrality gap instances for SA+SDP also hard for Lasserre SDP?Previous result [BBHKS

Z

'12]

No for

UG

8-round

Lasserre

sol

ves the

Unique Games

lowerbound

instancesSlide11

From SA+SDP to Lasserre SDP (cont’d)Are the integrality gap instances for SA+SDP also hard for Lasserre SDP?This paperNo for

Max-Cut

and

Balanced Separator

Constant-round

Lasserre

gives better-than-0.878 approximation for

Max-Cut

lowerbound

instances

4-round

Lasserre

gives con

stant approximation for the the Balanced Separator lowerbound instancesSlide12

Proof overview Integrality gap instanceSDP completeness: good vector solutionIntegral soundness:

no good integral

solution

Show the instance is not integrality gap instance for

Lasserre

SDP –

no good vector solution

we bound the value of the dual of the SDP

interpret the dual as a proof system

(”

SOS

d

/sum-of-squares proof system")

lift the soundness proof to the proof systemSlide13

What is the SOSd proof system?Slide14

Polynomial optimizationMaximize/MinimizeSubject to all functions are low-degree n-variate polynomials

Max-Cut example:

Maximize

s.t.

Slide15

Polynomial optimization (cont'd)Maximize/MinimizeSubject to all functions are low-degree n-variate polynomials

Balanced

Separator

example:

Minimize

s.t.

Slide16

Certifying no good solutionMaximizeSubject toTo certify that there is no solution better than , simply say that the following equalities & inequalities are infeasibleSlide17

The Sum-of-Squares proof systemTo show the following equalities & inequalities are infeasible,Show thatwhere is a sum of squared polynomials, including 's

A degree-d "Sum-of-Squares" refutation, whereSlide18

PositivstellensatzSubject to some mild technical conditions,every infeasible system has such a refutation

Caveat:

f

i

’s

and h might

need to have high degree.

Lasserre

SDP and

SOS

d

proof system

A degree-d SOS refutation



O(d)-round

Lasserre

SDP is infeasible Slide19

SummaryDefined the degree-d SOS proof systemRemaining task Integral soundness proof  low-degree refutation in the SOS proof systemSlide20

Example 1To refuteWe simply writeA degree-2 SOS

refutationSlide21

One-slide How-to

Thm

: Min-Balanced-Separator

in this graph is

≥ blah

Proof: …

hypercontractivity

…

“Check out these polynomials.”

Thm

: Max-Cut of this graph

is

≤ blah

Proof: … Invariance Principle …

… Majority-Is-

Stablest

…

“Check out these polynomials.”Slide22

Example 2: Max-Cut on triangle graphTo refuteWe "simply" write ... ...Slide23

Example 2: Max-Cut on triangle graph

(cont'd)

A degree-4 SoS refutation

Slide24

Latest resultsOur theorem on Max-Cut is improved by [DMN’12]Constant-round Lasserre SDP almost exactly solves the known instances

Constant-round

Lasserre

SDP solves the hard instances for Vertex-Cover

[KOT

Z

’12]

Open question

Does constant-round

Lasserre

SDP solve the known instances for all the CSPs?

I.e. SOS-

ize

Raghavendra’s

theorem.Slide25

Thank you!