Linear RoundIntegrality Gapsfor the

Lasserre Hierarchy

Grant Schoenebeck

Max Cut IP

}1,0{

2max),(

i

Ejijiji

xVi

xxxx

Given graph GPartition vertices into two sets toMaximize # edges crossing partition

Max Cut IP Homogenized

iii

Ejijiji

xxxxVi

x

xxxxxx

0

20

),(00

1

2max

Max Cut SDP [GW94]

2

0

2

0

),(00

,

1

,2,,max

ii

Ejijiji

vvvVi

v

vvvvvv

Integrality Gap = min

Integrality Gap = ) – Approximation AlgorithmIntegrality Gap ¸ .878… (rounding)[GW]Integrality Gap · .878… (bad instance) [FS]

Integral SolutionSDP Solution

Max Cut SDP

2

0

2

0

),(00

,

1

,2,,max

ii

Ejijiji

vvvVi

v

vvvvvv

884.0552.4

4

Solution SDP

Solution Integral Gapy Integralit

0

14

23

v0

v1

v4

v2

v3

Max Cut SDP and ▲ inequality

222

0

20

),(00

,,

1

2max

kjjiki

iii

Ejijiji

xxxxxxVkji

xxxxVi

x

xxxxxx

Max Cut SDP and ▲ inequality

222

2

0

2

0

),(00

,,

,

1

,2,,max

kjjiki

ii

Ejijiji

vvvvvvVkji

vvvVi

v

vvvvvv

SDP value of 5-cycle = 4 General Integrality Gap Remains 0.878…

[KV05]

Max Cut IP r-juntas Homogenized

gfgf

gfgf

V

Ejiji

xxx

xxxx

gf

gfgfrgfgf

x

x

''

21

),(

0

juntas- 1,01,0,,,

1

max

Max Cut Lasserre r-rounds

gfgf

gfgf

V

Ejiji

vvvgf

vvvvgfgf

rgfgf

v

v

0

,,

juntas- 1,01,0,,,

1

max

''

2

1

),(

2

CSP Maximization IP

gfgf

gfgf

V

cc

xxx

xxxx

gf

gfgfrgfgf

x

x

''

21

sconstraint

0

juntas- 1,01,0,,,

1

max

CSP Maximization Lasserre r-rounds SDP

gfgf

gfgf

V

cc

vvvgf

vvvvgfgf

rgfgf

v

v

0

,,

juntas- 1,01,0,,,

1

max

''

2

1

contraints

2

CSP Satisfaction IP

gfgf

gfgf

V

c

xxx

xxxx

gf

gfgfrgfgf

xc

x

''

21

0

juntas- 1,01,0,,,

1sconstraint

1

CSP SatisfiablityLasserre r-rounds SDP

gfgf

gfgf

V

c

vvvgf

vvvvgfgf

rgfgf

vc

v

0

,,

juntas- 1,01,0,,,

1sconstraint

1

''

2

2

1

Lasserre Facts Runs in time nr

Strength of Lasserre Tighter than other hieracheis

Serali-Adams Lavasz-Schrijver (LP and SDP)

r-rounds imply all valid constraints on r variables tight after n rounds

Few rounds often work well 1-round ) Lovasz -function 1-round ) Goemans-Williamson 3-rounds ) ARV sparsest cut 2-rounds ) MaxCut with ▲inequality

In general unknown and a great open question

Main Result

Theorem: Random 3XOR instance not refuted by (n) rounds of Lasserre

3XOR: =

0

1

0

651

743

721

xxx

xxx

xxx

Previous LS+ Results

3-SAT 7/8+ (n) LS+ rounds [AAT]Vertex Cover 7/6- 1 rounds [FO] 7/6- (n) LS+ rounds [STT] 2- (√log(n)/loglog(n)) LS+ rounds [GMPT]

LB for Random 3XOR

Theorem: Random 3XOR instance not refuted by (n) rounds of Lasserre

Proof: Random 3XOR cannot be refuted by

width-w resolutions for w = (n) [BW]

No width-w resolution ) no w/4-Lasserre refutation

Width w-Resolution

Combine if result has · w variables

1861 xxx 0876 xxx

171 xx

071 xx

10

Width w-Resolution

Combine if result has · w variables

1861 xxx 0876 xxx 071 xx

Idea / Proof ) width-2r Res ) F = linear functions “in” L(r) = linear function of r-variables

L1, L2 2 F Å ) L1 Δ L2 2 ξ=L(r)/F = {[Ø][L*

2], [L*2], …}

Good-PA = Partial assignment that satisfies ~ ,

for every Good-PA: = for every Good-PA:

1Δ1

0Δ1*

*

LL

LLL

Idea / Proof L(r) = linear function of r-variables F = linear functions in C ξ = L(r)/F = {[Ø][L*

2], [L*2], …}

C1Δ1

C0Δ1

LL

LLL

][][

)()(ˆs

nSf eSSfv

cv

eIev

eIev

x

c

Ix

Ix

IiIi

iIi

iIi

if ),0,0,1(2

1)(

2

12

1)(

2

12

1

2

1

][][1

][][0

)()(ˆ])([

][

SSfXvXS

f

gfgf

gfgf

V

c

vvvgf

vvvvgfgf

rgfgf

vc

v

0

,,

juntas- 1,01,0,,,

1sconstraint

1

''

2

2

1

Multiplication Check

][

][ ][

][ ][

][

)()(

)(ˆ)(ˆ)(

)(ˆ)()(ˆ)(

])([ˆ])([ˆ

,

Y

Y nX

nX Y

X

gf

YfgY

YXgXfY

YXgYXXfX

xgxf

vv

^

Corollaries

Meta-Corollary: Reductions easyThe (n) level of Lasserre: Cannot refute K-SAT IG of ½ + for Max-k-XOR IG of 1 – ½k + for Max-k-SAT IG of 7/6 + for Vertex Cover IG ½ + for UniformHGVertexCover IG any constant for

UniformHGIndependentSet

Pick random 3SAT formula Pretend it is a 3XOR formula

Use vectors from 3XOR SDP to satisfy 3SAT SDP

Corollary I

Random 3SAT instances not refuted by (n) rounds of Lasserre

1 kjikji xxxxxx

gfgf

gfgf

V

c

vvvgf

vvvvgfgf

rgfgf

vc

v

0

,,

juntas- 1,01,0,,,

1sconstraint

1

''

2

2

1

22

\

2

\

SATcSATcXORcXORc

SATcSATcXORcXORc

vvv

vvv

Corollary II, III

Integrality gap of ½ + ε after (n) rounds of Lasserre forRandom 3XOR instance

Integrality gap of 7/8 + ε after (n) rounds of Lasserre forRandom 3SAT instance

Vertex Cover Corollary

Integrality gap of 7/6 - ε after (n) rounds of Lasserre for Vertex Cover

FGLSS graphs from Random 3XOR formula (m = cn clauses)

(y1, …, yn) Lasr(VC) (1-y1, …, 1-yn) Lasr(IS)

Transformation previously constructed vectors

x1 + x2 + x3 = 1001

100111

010

x3 + x4 + x5 = 0

101

110

011

000

SDP Hierarchies from a Distance

Approximation Algorithms Unconditional Lower

Bounds Proof Complexity Local-Global Tradeoffs

Future Directions

Other Lasserre Integrality Gaps Positive Results Relationship to Resolution