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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