Tight integrality gaps forvertex-cover semidefinite relaxations
in the Lovász-Schrijver Hierarchy
Avner Magen
Joint work with Costis Georgiou, Toni Pitassi and Iannis Tourlakis
University of Toronto
Minimum Vertex Cover
Finding minimum size VC is NP-hard
Exist simple 2-approximations
All known algs are 2 o(1) approximations!
Probabilistically checkable proofs (PCPs) No poly-time 1.36 approximation [Dinur-Safra’02]
Unique Games Conjecture [Khot’02] No poly-time 2 approximation [Khot-Regev’03]
Alternative (concrete) approach [ABL’02, ABLT’06]: Rule out approximations by large subfamilies of algorithms
Linear Programming approach
min iV vi
vi + vj ≥ 1, ij E
vi {0,1} 0 ≤ vi ≤ 1
True Optimum
Optimal Fractional SolutionIntegrality Gap: max
Easy to see IG ≤ 2
for Kn : IG = 2 1/n
SDP: the ultimate remedy?
Vertex Cover on G = (V,E)
Tighter relaxation? Smaller integrality gap?
min iV (1 + v0 · vi)/2
(v0 vi) · (v0 vj) = 0, ij E
|| vi ||2 = 1, vi Rn+1
min iV (1 + x0xi)/2
(x0 xi)(x0 xj) = 0, ij E
|xi| = 1 Hatami-M-Markakis’06:
Integrality gap still 2 o(1), even
with “pentagonal” inequalities
Semidefinite Programming Relaxations
Kleinberg-Goemans’98:
Integrality gap 2 o(1)
Clearly holds in
integral case
vi {1,1}
(v0 vi) · (v0 vj) 0, i,j
(vi vj) · (vi vk) 0, i,j
Charikar’02:
Gap still 2 o(1)
Systematic Approach: Lovász-Schrijver Liftings [LS’91]
Procedures LS0, LS, LS+ for tightening linear relaxations Integral hull in ≤ n rounds Optimize over rth round relaxation in nO(r) time
Very powerful algorithms obtained through small number of rounds: GW’94, KZ’97, ARV’04 algorithms “poly-time” in LS+ All NP in “exponential time”
May view super-constant rounds lower bounds in LS+ models as evidence about inapproximability
Initial Linear Relaxation
Integral Hull
Has PSD constraint Sequence of tighter and tighter SDPs
“Lift” to obtain
SDP Relaxation
n variablesn2 variables
“Project” back
to obtain tighter LP
Previous Lower Bounds for Vertex Cover – without SDP constraints (LS)
[ABLT’06]: Int. gap 2 o(1) after (log n) LS rounds
[Tourlakis’06]: Int. gap 1.5 o(1) after (log2 n) LS rounds
[STT’06b]: Int. gap 2 o(1) after (n) LS rounds
Status of SDP variant LS+
Stronger: one round already Implies clique constraint More generally, gives n-θ(G) lower bound on VC (so
sparse graph are generally not good) Gives rise to SDPs in the “lift” phases.
Integrality gap of 7/6 for LS+ (STT06a)
PCP world: Hastad 0.5-hardness for MAX3XOR and the FGLSS reduction imply 7/6-hardness for VC
AAT05 proved matching LB (for int. gap) in LS+ world for MAX3XOR
STT06b using further ideas from FO06, extend AAT MAX3XOR LB to prove 7/6 int. gap for linear rounds
graph family: FGLSS reduction on random MAX3XOR instances Int. gap 7/6 already after one round
Vertex Cover in LS: results so far
SDP version (LS+)?
Int. gap ≥
2-o(1) ?# rounds
superconsant?
ABLT ’02,STT ’07 NO YES YES
STT ’06 YES NO YES
Charikar ’02 YES YES NO
New result YES YES YES
Main Result
Theorem: Int. gap 2 o(1) for SDPs resulting after(√log n/log log n) LS+ rounds
One LS+ round tighter than [C’02] SDP
SDPs ruled out incomparable to SDPs with (generalized) triangle and pentagonal inequalities (e.g., [HMM’06])
Theorem: Int. gap 2 O(1/√log n/log log n)after O(1) LS+ rounds
Karakostas [K’05] SDP gives 2 (1/√log n) approximation
Use same graph families as [KG’98], [C’02], [HMM’06]SDP solutions rely on sequence of polynomials applying
tensor operations on vectors
xk(xi + xj x0) 0 ij E (x0 xi)(xj – x0) 0 ij E(x0 xi)(x0 xj) 0
vk · (vi + vj v0) 0 ij E(v0 vi) · (v0 vj) = 0 ij E(v0 vi) · (v0 vj) 0
xk(xi + xj x0) 0 ij E (x0 xi)(x0 – xj) = 0 ij E(x0 xi)(x0 xj) 0
Yik + Yjk Y0k 0 ij EY00 Y0i Y0j + Yij = 0 ij EY00 Y0i Y0j + Yij 0
Convert vertex cover LP into an SDP?
Multiply linear inequalities to get valid quadratic constraints.Crucially, add integrality conditions: (x0 xi)xi = 0
E.g.,
Linearize: replace products xixj with linear variables Yij
Lifted SDP in (n + 1)2 variablesProject resulting convex body back onto n + 1 variables Y0i
xk(xi + xj x0) 0 ij E (x0 xi)(xi + xj x0) 0 ij E (x0 xi)(x0 xj) 0
LS+ lift-and-project: the quick guide
min iV xi
xi + xj 1 (i,j) E0 xi 1 i V(x0 = 1)xi + xj x0 0 (i,j) E xi 0 i V x0 xi 0 i V
Yei ,Y(e0ei) K
Y0i = Yii
(x0 xi)xi = 0
(x0 = 1)
Y is PSD
Homogenization:
cone K
= xi
How LS and LS+ tighten VC Relaxation
One round of LS precisely adds “odd-cycle constraints”: For all cycles C in G of odd length,
iC xi ≥ (|C|+1)/2
x1 + x2 + x3 ≥ 2
One round of LS+ adds more: Clique constraints: For all cliques K in G,
iK xi ≥ |K| – 1
min iV xi
xi + xj ≥ 1, ij E
0 ≤ xi ≤ 1
vs. x1 + x2 + x3 ≥ 3/2
Deriving the clique constraints in LS+
0 ≤ x0 – xi) (xi + xj – x0) +((k –x0 – xi)i
2
Edge constraint
i≠j
Let K be a clique of size k in G
SDP condition
xi2 – (k – 1) x0
2
xi ≥ k – 1After projectingi
x K(r) if matrix Y s.t. diagonal is x Y is PSD “columns” K(r 1)
Proving Lower Bounds in LS+ Hierarchies
I.H.
LP relaxation K for G with min VC ~ n:
xi + xj ≥1 ij E
(½, ½,…)
K(1) K(3)
K(2)
Int. gap of K is ≥ 2 – o(1)
(½+, ½+ …)
Use inductive proof: find appropriate Y’s
“Protection”
matrix for xLemma (LS’91):
“Frankl-Rödl” graphs
m-dimensional Hamming cube: n = 2m points
V = {1,1}m
(i, j) E iff (i, j) = (1 )m }
parameter
Theorem: [Frankl-Rödl’87]
Max Ind.Set size |B(v,n/2(1- ))|
m2m(1 2/64)m
Cor: If = (√log m/m) then max IS is o(2m) = o(n)
Graphs used for int.gaps in [KK91, AK94, KG95, C02, HMM06]
(i, j) = (1 )m
o(n)
What’s so wonderful about them?...
Start with a perfect matching
Perturb : edges connect
vertices of Ham. Dist. (1-)n
Vertex Cover = n/2
``Geometric’’ vertex cover = n/2 +O( )
Proof Outline
In induction: need vectors vi to define matrix Yij = vi vj
Show vi exist whenever x {0, 1, ½ + }n and > 6
Ensure S {0, 1, ½ + }n where O()
(/) round lower bound for x = (½ + )1Constant and = (√log m/m)
Int. gap 2 o(1) after (√log n/log log n) rounds
x K(r) if PSD matrix Y s.t.
1. diagonal is x
2. “columns” K(r1) 2’. Show some set S K(r1)
where “columns” conv(S)
(i, j) = (1 )m
VC 1 o(n)
x = (½ + )1
Back to Frankl-Rödl graphs
Natural set {ui} of unit vectors: {1,1}m
(v0 vi) · (v0 vj) = 0, (i, j) E
√m1
Note: ui · uj = 1 2(i, j)/m
Hence (i, j) E ui and uj nearly antipodal
Nearly true for vi = ui
21 for (i, j) E
linear function
F of vi · vj
(i, j) = (1 )m
VC 1 o(n)
ui · uj
1
21
1
F
1
0
1
vi · vj
1
1
Kleinberg-Goemans:
Affine translation onui to obtain vi
F
V = {1,1}m
Use Kleinberg-Goemans vi for LS+?
Fact: One round of LS+ also requires following ineq:
Idea (Charikar): Map ui to wi s.t.
F(wi · wj) 0
F(wi · wj) = 0 if ij E
I.e, when ui · uj = 2 1
How? Use tensoring
(v0 vi) · (v0 vj) 0 i,j
equality whenever ij E
(i, j) = (1 )m
VC 1 o(n)ui · uj
1
21
1
F(vi · vj)1
0
1
vi · vj
1
1
[KG] affine
map on ui
linear
map
F(vi · vj)
F(wi · wj)1
0
1
Desired mapping
on dot-products
Tensoring
u, v Rn
Tensor product: u v Rn2
Value uivj at coordinate (i, j) [n]2
Easy fact: (u v) · (u v) = (u · v)2
Let P(x) = c1xt1 + … + cqxtq
Consider map TP(u) = (c1ut1,…, cqutq)
Example: P(x) = x2 + 4x TP(u) = (u u, 2u) Rn2+2n
TP(u) · TP(v) = (u · v)2 + 4(u · v)2 = P(u · v)
Fact: TP(u) · TP(v) = P(u · v)
22
P determines dot-product
of resulting vectors
Positive coefficients
Back to finding solution for stronger SDP: Use TP
Charikar exhibits appropriate P
(i, j) = (1 )m
VC 1 o(n)
I.e, when ui · uj = 2 1
(v0 vi) · (v0 vj) 0 i,j
equality whenever (i, j) E
F(vi · vj)
ui · uj
1
21
1F
1
0
1
Want wi = TP(ui)s.t. F(wi · wj) min at (i, j) E
ui · uj 11
21
0
KG
C
Charikar sol’n gives one round LS+ lower bound
Charikar vectors define Yij = vi · vj that:
n Diagonal is x = (½ + )1n “Columns” K
x K(r) if PSD matrix Y s.t.
1. diagonal is x
2. “columns” K(r1) I.H.
x = (½ + )1
Can Charikar vectors show “columns” K(1)?
VC = 1 o(n)
Must have seq
of polynomials
Problems: (1) “Columns” not of form (½ + )1 (2) Charikar’s vectors work only for one value
Values distributed like
polynomial of Gaussian
Making non-uniform “columns” uniform
“Columns” we want to continue from not of form (½ + )1
Def [STT]: x K is -saturated if for all ij E so that xi, xj < 1 there is surplus: xi + xj 1 + 2
Lemma [STT]: x is -saturated there exists set of vectorsx(i) {0, 1, ½ + }n in K s.t. x conv({x(i) }).
Can convert “columns” to (essentially) (½ + )1IF “columns” are -saturated
Will be safe to “ignore” 0/1
values distributed like polynomial of Gaussian
Goal: matrix Y for x with “column” saturation ()
Recall P(x) defines TP(u) such that TP(u) · TP(v) = P(u · v)
deg(PC) = O(1/)
Fact: Y has “columns” s.t. some edges never have surplus
Problem: saturation of “close by” edges?
Saturation
Normal. Ham. Dist. from blue edge
Necessary: deg(P) ≥ · m !
For all P
P
Bad saturation zone
The blue edge
~ P(1)P(1-1/m)
≤ P’(1)/m
Is saturation good enough?
= o(m)
Want column saturation O()
Precise technical property needed for P:
| P(ui · uk ) + P(uj · uk) | O()
For all vertices k and all edges ij :
[1, 1]
But ui · uj = 2 1 for all edges ij, so
Need | P(x) + P(y) | O() over R
Red points correspond to 0-1 edges Ignored in saturation calculation
1
1
1
1
12
12
21
21
R
11/m
11/m x
y
Domain of P(x) + P(y)
|ui·uk+uj·uk| 2
|ui·ukuj·uk| 2(1-)
So far: There must be a seq of polys dep. on m. Polynomials must have large degree.
Let x {0, 1, ½ + }n
Take P(x) = (xx 1)m/ + x 1/ + (1- x
Properties: Minimum at ui · uj, ij E P’(1) > m Works as long as > 6 The “Columns” of Y that is produced by
using TP,m(ui) have saturation O()
ui · uj 11 21 0
KG
C
P
arbitrary > 6
Defining the sequence of tensoring polynomials
Putting everything together
Induction: Have x {0, 1, ½ + }n where > 6
Define Y using TP,m(ui)
“Columns” have saturation O()
[STT] Exists S K {0, 1, ½ + }n s.t. “columns” conv(S)
Induction Hypothesis S K(r 1)
Take constant and = (√log m/m)
x K(r) if PSD matrix Y s.t.
1. diagonal is x
2. “columns” K(r1) 2’. Show some set S K(r1)
where “columns” conv(S)
x = (½ + )1
r = (/)
(i, j) = (1 )m
VC 1 o(n)
x K(r)
Requiring that ||vi-vj||2 is l1?
As is, no l1 inequalities are not implied. The results of [HMM] (showing that metric-cut ineqaities
and pentagonal inequalities hold) suggest the examples are still good.
Need to Give Sherali Adams LB introduce dij = ||vi-vj||2
Add more reqs the LS+ proof need to satisfy.
Sherali-Adams [SA’90] Lift-and-Project
Idea: Keep “lifting” but never project!Simulate third, fourth, etc, degree products with linear vars
Only known integrality gap [FK’06]:(log n) SA rounds int. gap ≤ 2 for MAX-CUT
SA+ lower bound would inequalities for lifted variables Triangle, pentagonal, etc., inequalities derivable
E.g., x1x2x3 Y123
LP not SDP version
Relations to Unique Games Conjecture (UGC)
LS+ lower bounds may provide evidence of inapproximability
UGC [Khot’02] implies optimal inapproximability results for Vertex Cover, MAX-CUT, etc
Strong LS+, SA+ lower bounds for VC, MAX-CUT
Thanks
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