Optimal Algorithms and
Inapproximability Results for
Every CSP?
Prasad Raghavendra
University of Washington
Seattle
Constraint Satisfaction Problem
A Classic Example : Max-3-SAT
( x1  x 2  x3 )( x2  x3  x5 )( x 2  x3  x5 )( x5  x4  x1 )
Equivalently the
largest fraction of
clauses
Given a 3-SAT formula,
Find an assignment to the variables that
satisfies the maximum number of clauses.
Constraint Satisfaction Problem
Problem :
Domain :
Predicates :
Pi :
Instance :
Max-3-SAT
{0,1,.. q-1}
{P1, P2 , P3 … Pr}
[q]k
Domain :
Predicates :
{0,1}
P1(x,y,z) = x ѵ y ѵ z
-> {0,1}
( x1  x 2  x3 )( x2  x3  x5 )( x 2  x3  x5 )( x5  x4  x1 )
• Set of variables.
• Predicates Pi applied on variables
Find an assignment that satisfies the
largest fraction of constraints.
Variables :
{x1 , x2 , x3 ,x4 , x5}
Constraints : 4 clauses
Generalized CSP (GCSP)
Replace Predicates by Payoff Functions
(bounded real valued)
Problem :
Domain :
Pay Offs:
Objective :
{0,1,.. q-1}
{P1, P2 , P3 … Pr}
Pi : [q]k -> [-1, 1]
Pay Off Functions can
be Negative
Find an assignment
that maximizes the
Average Payoff
Can model Minimization Problems
like Multiway Cut, Min-Uncut.
Examples of GCSPs
Max-3-SAT
Max Cut
Max Di Cut
Multiway Cut
Metric Labelling
0-Extension
Unique Games
d- to - 1 Games
Label Cover
Horn Sat
Unique Games
A Special Case
x-y = 11 (mod 17)
x-z = 13 (mod 17)
…
….
z-w = 15(mod 17)
E2LIN mod p
Given a set of linear equations of the
form:
Xi – Xj = cij mod p
Find a solution that satisfies the
maximum number of equations.
Unique Games Conjecture [Khot 02]
An Equivalent Version
[Khot-Kindler-Mossel-O’Donnell]
For every ε> 0, the following problem is
NP-hard for large enough prime p
Given a E2LIN mod p system, distinguish
between:
• There is an assignment satisfying 1-ε fraction
of the equations.
• No assignment satisfies more than ε fraction
of equations.
Unique Games Conjecture
A notorious open problem, no general consensus
either way.
Algorithm
[Khot 02]
On (1-Є) satisfiable instances
1  O( p 2 1/ 5 log( 1 /  ) )
[Trevisan]
1  O(3  log n )
[Gupta-Talwar]
1 – O(ε logn)
[Charikar-Makarychev-Makarychev]
[Chlamtac-Makarychev-Makarychev]
[Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi]
p  /(2 )
1  O( log n log p )

1
1 log


Hardness Results:
No constant factor approximation for unique games. [FeigeReichman]
Why is UGC important?
Problem
Vertex Cover
Max CUT
Max 2- SAT
SPARSEST CUT
Max k-CSP
Best
Approximation
Algorithm
2
0.878
0.9401
log n

 k / 2k

NP Hardness
1.36
0.941
0.9546
1+ε
O 2 k / 2k


Unique Games
Hardness
2
0.878
0.9401
Every Constant
O k / 2k
UG hardness results are intimately
connected to the limitations of
Semidefinite Programming


Semidefinite
Programming
Max Cut
Input : a weighted
graph G
10
15
7
1
1
3
Find a cut that
maximizes the number
of crossing edges
Max Cut SDP
Quadratic Program
Semidefinite
Program
-1
10
1
15
-1
1
7
1
1
1
3
-1
-1
-1
-1
Variables : vx1 , xv2 … xvn
x|i =vi1|2or= -1
1
Maximize
1
2
w
(
|
x
v

v
x
)
|

ij
ij
ii
j
4 (i , j )E
Relax all the xi to be unit vectors instead of {1,-1}.
All products are replaced by inner products of vectors
MaxCut Rounding
v2
v1
v3
Cut the sphere by a random
hyperplane, and output the
induced graph cut.
- A 0.878 approximation for
the problem.
v5
v4
General Boolean 2-CSPs
In Integral Solution
vi = 1 or -1
V0 = 1
Total PayOff
Triangle Inequality
2-CSP over {0,..q-1}
Total PayOff
Arbitrary k-ary GCSP
•SDP is similar to the one used by [Karloff-Zwick]
Max-3-SAT algorithm.
•It is weaker than k-rounds of Lasserre / LS+
heirarchies
Results
Fix a GCSP
Two Curves
Integrality Gap Curve
If UGC is false:
true:
U(c)
U(c)≥ S(c)
is
meaningless!
S(c) = smallest value of
the integral solution,
given SDP value c.
UGC Hardness Curve
U(c)
S(c)
0
Optimum Solution
1
U(c) = The best polytime
computable solution,
assuming UGC given an
instance with value c.
UG Hardness Result
Theorem 1:
For every constant η > 0,
and every GCSP Problem,
U(c) < S(c+ η) + η
U(c)
U(c)
S(c)
0
Optimum Solution
c
= SDP Value
S(c) = SDP Integrality Gap
U(c) = UGC Hardness Curve
1
Roughly speaking,
Assuming UGC, the SDP(I), SDP(II),SDP(III) give best
possible approximation for every CSP
Consequences
If UGC is true, then adding more constraints
does not help for any CSP
Lovasz-Schriver, Lasserre, Sherali-Adams
heirarchies do not yield better approximation
ratios for any CSP in the worst case.
Efficient Rounding
Scheme
U(c)
S(c)
0
Optimum Solution
1
c
= SDP Value
S(c) = SDP Integrality Gap
U(c) = UGC Hardness Curve
Roughly speaking,
There is a generic
polytime rounding
scheme that is optimal for
every CSP, assuming UGC.
Theorem:
For every constant η > 0, and every GCSP,
there is a polytime rounding scheme that
outputs a solution of value U(c-η) – η
If UGC is true, then for every Generalized
Constraint Satisfaction Problem :
If UGC is false??
NP-hard
•Hardness result
doesn’t make sense.
U(c)
S(c)
algorithm
0
Optimum Solution
1
•How good is the
rounding scheme?
S(c) = SDP Integrality Gap
Unconditionally
Roughly Speaking,
For 2-CSPs, the Approximation ratio obtained is at least the
red curve S(c)
The rounding scheme achieves the integrality gap of SDP for
2-CSPs (both binary and q-ary cases)
Theorem:
Let A(c) be rounding scheme’s
performance on input with
SDP value = c. For every
constant η > 0
S(c)
0
Optimum Solution
1
A(c) > S(c- η) - η
As good as the best
SDP(II) and SDP(III) are the
strongest SDPs used in
approximation algorithms
for 2-CSPs
The Generic
Algorithm is at least
as good as the best
known algorithms
for 2-CSPs
Examples:
Max Cut
Max-2-SAT
Unique Games
[Goemans-Williamson]
[Lewin-Livnat-Zwick]
[Charikar-Makarychev-Makarychev]
Computing Integrality Gaps
S(c)
0
Theorem:
For any η, and any 2-CSP, the
curve S(c) can be computed
within error η.
(Time taken depends on η and
domain size q)
Optimum Solution 1
Explicit bounds on the size of an
integrality gap instance for any 2-CSP.
Related Work
Problem
Vertex Cover
Max CUT
Max 2- SAT
SPARSEST CUT
Max k-CSP
Best
Approximation
Algorithm
2
0.878
0.9401
log n
 k / 2k


Unique Games Hardness
2
[Khot-Regev]
0.878
[Khot-Kindler-Mossel-O’donnell]
0.9401
[Per Austrin]
Every Constant [Chawla-Krauthgamer-..]
[Trevisan-Samorodnitsky]
O k / 2k


[Austrin 07]
Assuming UGC, and a certain additional conjecture:
``For every boolean 2-CSP, the best approximation
is given by SDP(III)”
[O’Donnell-Wu 08]
Obtain matching approximation algorithm, UGC hardness
and SDP gaps for MaxCut
Proof Overview
Dictatorship Test
F is a dictator function
F(x1 ,… xR) = xi
Pr[ACCEPT ] =
Completeness
Given a function
F : {-1,1}R
{-1,1}
•Toss random coins
•Make a few queries to F
•Output either ACCEPT or
REJECT
F is far from every
dictator function
(No influential coordinate)
Pr[ACCEPT ] =
Soundness
Connections
SDP Gap
Instance
SDP = 0.9
OPT = 0.7
[Khot-Vishnoi]
For sparsest cut, max cut.
[This Paper]
UG
Hardness
Dictatorship
Test
0.9 vs 0.7
Completeness = 0.9
Soundness = 0.7
[Khot-Kindler-Mossel-O’Donnell]
All these conversions hold for every GCSP
A Dictatorship Test for Maxcut
A dictatorship test is a graph
G on the hypercube.
A cut gives a function F on the
hypercube
Completeness
Value of Dictator Cuts
F(x) = xi
Hypercube = {-1,1}100
Soundness
The maximum value
attained by a cut far from
a dictator
v2
10
15
1
1
7
v1
v3
3
Graph G
v5
v4
SDP Solution
An Example : Maxcut
Completeness
Value of Dictator Cuts =
SDP Value (G)
Soundness
Given a cut far from every
dictator :
It gives a cut on graph G
with the same value.
100 dimensional hypercube
In other words,
Soundness ≤ OPT(G)
From Graphs to Tests
10
15
7
1
1
3
Graph G (n vertices)
H
v2
v1
v3
Constant
independent of
size of G
100 dimensional hypercube : {-1,1}100
v5
v4
SDP Solution
For each edge e, connect every pair of
vertices in hypercube separated by
the length of e
-1
v2
v1
v3
-1
Completeness
-1
v5
1
v4
1
1
100 dimensional hypercube
Echoice of edge e=(u,v) in G
For each edge e, connect
every pair of vertices in
hypercube separated by the
length of e
Set F(X) = X1
[EX,Y in 100 dim hypercube with dist |u-v|^2 [
(F(X)-F(Y))
(X1 – Y1)2 2 ]
X1 is not equal to Y1 with probability |u-v|2 ,
hence completeness = SDP Value (G)
]
The Invariance Principle
Invariance Principle for Low Degree Polynomials
[Rotar] [Mossel-O’Donnell-Oleszkiewich], [Mossel 2008]
“If a low degree polynomial F has no influential
coordinate, then F({-1,1}n) and F(Gaussian) have
similar distribution.”
A generalization of the following fact :
``Sum of large number of {-1,1} random variables
has similar distribution as
Sum of large number of Gaussian random variables.”
From Hypercube to the Sphere
F :
100
Dimensional
hypercube
[-1,1]
Express F as a multilinear polynomial using Fourier expansion, thus
extending it to the sphere.
P :
100
dimensio
nal
sphere
Nearly
Real
numbers
always
[-1,1]
Since F is far from a dictator, by invariance principle, its behaviour on the sphere is
similar to its behaviour on hypercube.
A Graph on the Sphere
10
15
7
1
1
3
Graph G (n vertices)
S
v2
v1
v3
100 dimensional sphere
v5
v4
SDP Solution
For each edge e, connect every pair of
vertices in sphere separated by the
length of e
Hypercube vs Sphere
H
S
P : sphere -> Nearly {-1,1}
F:{-1,1}100
-> {-1,1}
is a cut far from every
dictator.
Is the multilinear extension of
F
By Invariance Principle,
MaxCut value of F on H
≈
Maxcut value of P on S.
v2
v1
Soundness
v3
v5
G
v4
S
For each edge e in the graph G
connect every pair of vertices
in hypercube separated by
the length of e
Alternatively, generate S as follows:
Take the union of all possible rotations of the graph G
S consists of union of disjoint copies of G. Thus,
MaxCut Value of S < Max cut value of G.
Hence MaxCut value of F on H is at most the max cut value of G.
Soundness ≤ MaxCut(G)
Algorithmically,
Given a cut F of the hypercube graph H
• Extend F to a function P on the sphere using
its Fourier expansion.
• Pick a random rotation of the SDP solution to
the graph G
• This gives a random copy Gc of G inside the
sphere graph S
• Output the solution assigned by P to GC
Given the Polynomial P(y1,… y100)
Roughly
Formally
Sample R Random Sample R independent vectors : g(1), g(2) ,.. g(100)
Directions
Each with i.i.d Gaussian components.
Project each vi along all directions g(1), g(2) ,..
Project along
g(100)
them
Yi(j) = v0∙vi + (1-ε)(vi – (v0∙vi)v0) ∙ g(100)
Compute P on
projections
Compute
xi = P(Yi(1) , Yi(2) ,.. Yi(100))
Round the output If xi > 1,
xi = 1
of P
If xi < -1,
xi = -1
If xi is in [-1,1]
xi = 1 with probability (1+xi)/2
-1 with probability (1-xi)/2
Key Lemma
1) Tests of the verifier are same as
the constraints in instance G
2) Completeness = SDP(G)
DICTG
Any
CSP Instance
G
Dictatorship Test
on functions
F : {-1,1}n ->{-1,1}
Any
Function
F: {-1,1}n → {-1,1}
RoundF
If F is far from a dictator,
RoundF (G)
Rounding Scheme
on CSP Instances G
≈ DICTG (F)
UG Hardness Result
Worst Case
Gap Instance
Dictatorship
Test
Completeness = c
Soundness <= s
Instance
SDP = c
OPT = s
UG
Hardness
Completeness = c
Soundness <= s
Theorem 1:
For every constant η > 0, and every GCSP Problem,
U(c) < S(c+ η) + η
Generic Rounding
Scheme
Solve SDP(III) to obtain vectors (v1 ,v2 ,… vn )
Add little noise to SDP solution (v1 ,v2 ,… vn )

For all multlinear polynomials
P(y1 ,y2, .. y100)
do
Round using P(y1 ,y2, .. y100)
Output the best solution obtained
P is Multilinear polynomial in 100 variables with coefficients in [-1,1]
Algorithm
Soundness of any Dictatorship Test
≥ U(c)
Any
Dictatorship
Dictatorship
Test (I)
Test = c
Completeness
UG
Hardness
Completeness = c
Completeness = c
Instance I
SDP = c
OPT = ?
There is some function F : {0,1}R -> {0,1} that has
Pr[F is accepted] ≥ U(c)
By Key Lemma, Performance of F as rounding
polynomial on instance I = Pr[F is accepted] > U(c)
Related Developments
• Multiway Cut and Metric Labelling problems.
[Manokaran, Naor, Schwartz, Raghavendra]
• Maximum Acyclic Subgraph problem
[Guruswami,Manokaran, Raghavendra]
• Bipartite Quadratic Optimization Problem
(Computing the Grothendieck constant)
[Raghavendra,Steurer]
Conclusions
Unique Games and Invariance Principle
connect :
Integrality Gaps, Hardness Results
,Dictatorship tests and Rounding Algorithms.
These connections lead to new algorithms,
and hardness results unifying several known
results.
Thank You
Rounding Scheme
(For Boolean CSPs)
Rounding Scheme was discovered by the
reversing the soundness analysis.
This fact was independently observed by Yi Wu
MaxCut Rounding
Cut the sphere by a random
hyperplane, and output the
induced graph cut.
v2
v1
v3
v5
v4
Equivalently,
•Pick a random direction g.
•For each vector vi , project
vi along g
yi = vi . g
•Assign
xi = 1 if yi > 0
xi = 0 otherwise.
SDP Rounding Schemes
SDP Vectors
(v1 , v2 .. vn )
Random Projection
Projections
(y1 , y2 .. yn )
For any CSP, it is enough to do
the following:
Instead of one random
projection, pick sufficiently
many (say 100) projections
Process the projections Use
a multi linear polynomial P
to process the projections
Assignment
UG Hardness Results
Worst Case
Gap Instance
Dictatorship
Test
Completeness = c
Soundness <= s
Instance
SDP = c
OPT = s
UG
Hardness
Completeness = c
Soundness <= s
Theorem 1:
For every constant η > 0, and every GCSP Problem,
U(c) < S(c+ η) + η
Multiway Cut and Labelling Problems
[Manokaran, Naor, Schwartz, Raghavendra]
10
15
7
1
1
3
3-Way Cut:
Separate the 3-terminals while
separating the minimum
number of edges
Theorem:
Assuming Unique Games
Conjecture,
The earthmover linear
program gives the best
approximation.
Theorem:
Unconditionally, the simple
SDP does not give better
approximations than the LP.
Maximum Acyclic Subgraph
[Guruswami,Manokaran, Raghavendra]
Theorem:
Assuming Unique Games
Conjecture,
The best algorithm’s output is
as good as a random
ordering.
Given a directed graph,
order the vertices to
maximize the number of
forward edges.
Theorem:
Unconditionally, the simple
SDP does not give better
approximations than random.
The Grothendieck Constant
[Raghavendra,Steurer]
The Grothendieck constant is the smallest constant
k(H) for which the following inequality holds for all
matrices :
The constant is just the integrality gap of the SDP for
bipartite quadratic optimization.
Value of the constant is between 1.6 and 1.7 but is
unknown yet.
Grothendieck Constant
[Raghavendra,Steurer]
Theorem:
There is an algorithm to compute arbitrarily good
approximations to the Grothendieck constant.
Theorem:
There is an efficient algorithm that solves the bipartite
quadratic optimization problem to an approximation equal
to Grothendieck constant.
If all this looks deceptively simple,
then it is because there was
deception
Working with several probability distributions at
once.
UG Hardness Results
Worst Case
Gap Instance
Dictatorship
Test
Completeness = c
Soundness <= s
Instance
SDP = c
OPT = s
Best UG Hardness
=
Integrality Gap
U(c) < S(c+η) + η
UG
Hardness
Completeness = c
Soundness <= s
Algorithm
Soundness of any Dictatorship Test
≥ U(c)
Any
Dictatorship
Dictatorship
Test (I)
Test = c
Completeness
UG
Hardness
Completeness = c
Completeness = c
Instance I
SDP = c
OPT = ?
There is some function F : {0,1}R -> {0,1} that has
Pr[F is accepted] ≥ U(c)
By Key Lemma, Performance of F as rounding
polynomial on instance I = Pr[F is accepted] > U(c)
Unconditional Results For 2-CSPs
On some instance I with SDP value = c ,
algorithm outputs a solution with value s.
For every function F far from dictator ,
Performance of F in rounding I
≤ s
By Key Lemma, For every such F
Pr[ F is accepted by Dict(I) ] ≤ s
Thus the Dict(I) is a test with soundness s.
Unconditional Results For 2-CSPs
Algorithm’s performance
matches the integrality
gap of the SDP
Dictatorship
Test(I)
Completeness = c
Soundness = s
Integrality Gap instance
SDP = c
OPT ≤ s
UG
Hardness
Completeness = c
Soundness = s
[Khot-Vishnoi]
UG Integrality
Gap instance
Computing Integrality Gaps
Integrality gap of a SDP relaxation
=
Worst case ratio of
Integral Optimum
SDP Optimum
Worst Case over all instances - an infinite set
Due to tight relation of integrality gaps/ dictatorship tests for 2-CSPs
Integrality gap of a SDP relaxation
=
Worst case ratio of
Soundness
Completeness
This time the worst case is along all dictatorship tests on {-1,1}R
- a finite set that can be discretized.
Key Lemma : Through An Example
SDP:
Variables : v1 , v2 ,v3
|v1|2 = |v2|2 = |v3|2 =1
1
2
3
Maximize

1
| v1  v2 |2  | v2  v3 |2  | v3  v1 |2
3

Local Random Variables
c = SDP Value
v1 , v2 , v3 = SDP Vectors
Fix an edge e = (1,2).
1
A13
A12
2
A23
3
There exists random
variables a1 a2 taking
values {-1,1} such that:
For every edge, E[a
thereais ]a local
distribution
over
=
v
∙
v
1 2
1 2
integral solutions such that:
All the moments
of order
at most2 2 match the
2
2
E[a1 ] = |v1|
E[a2 ] = |v2|2
inner products.
Dictatorship Test
Pick an edge (i,j)
Generate ai,aj in {-1,1}R as
follows:
The kth coordinates aik ,ajk
come from distribution Aij
Add noise to ai,aj
Accept if
F(ai) ≠ F(aj)
c =
SDP Value
v1 , v2 , v3 =
SDP Vectors
A12,A23,A31 = Local Distributions
Input Function:
F : {-1,1}R -> {-1,1}
1
Max Cut Instance
A12
2
3
Analysis
Pick an edge (i,j)
Generate ai,aj in {-1,1}R as follows:
The kth coordinates aik,ajk come
from distribution Aij
Add noise to ai,aj
Accept if
F(ai) ≠ F(aj)
A12,A23,A31 = Local Distributions
Max Cut Instance
1
2
3
Input Function:
F : {-1,1}R -> {-1,1}
1 1
1
1
2
2
2 
E A12 [( F (a1 )  F (a2 )) ]  E A23 [( F (a2 )  F (a3 )) ]  E A31 [( F (a3 )  F (a1 )) ]

3 4
4
4

A12,A23,A31 = Local Distributions
Completeness
Input Function is a Dictator :
F(x) = x1
1  11  1
2 2 11
2 2 
2 2 1 1
E A12 [(EFA12(a[(1 )a11F (aa2 ))21 )] ]  E AE
[( F[((aa221) Fa(31a3)))] ] EE A3131[([(Fa(31a3) a11F)(a]1 )) ]
23 A23


 
3  43  4
44
44
Suppose (a1 ,a2) is sampled from A12 then :
E[a11 a21] = v1∙ v2
E[a112] = |v1|2
E[a212] = |v2|2
EA12 [(a1  a2 ) ] | v1  v2 |
2
2
Summing up, Pr[Accept] = SDP Value(v1 , v2 ,v3)
Global Random Variables
g = random Gaussian vector.
(each coordinate generated by
i.i.d normal variable)
1
B
3
2
c = SDP Value
v1 , v2 , v3 = SDP Vectors
b1 = v1 ∙ g
b2 = v2 ∙ g
b3 = v3 ∙ g
There
is =a global
distribution
B=(b
,b
,bb3)]over
real
1
2
E[b
b
]
v
∙
v
E[b
b
]
=
v
∙
v
E[b
=
v
1 2
1 2
2 3
2 3
3 1
3∙ v1
numbers such that:
All the
moments
of order
at most
2 match
the
2
2
2
2
2
E[b1 ] = |v1| E[b2 ] = |v2| E[b3 ] = |v3|2
inner products.
Rounding with Polynomials
Input Polynomial : F(x1 ,x2 ,.. xR)
2
Generate
b1 = (b11 ,b12 ,… b1R)
b2 = (b21 ,b22 ,… b2R)
b3 = (b31 ,b32 ,… b3R)
with each coordinate (b1t ,b2t ,b3t) according to global
distribution B
1
B
3
Compute F(b1),F(b2) ,F(b3)
Round
F(b1),F(b2),F(b3) to {-1,1}
Output the rounded solution.
1 1
1
1
2
2
2 
EB [( F (b1 )  F (b2 )) ]  EB [( F (b2 )  F (b3 )) ]  EB [( F (b3 )  F (b1 )) ]

3 4
4
4

Invariance
Suppose F is far from every dictator then since A12
and B have same first two moments,
F(a1),F(a2) has nearly same distribution as
F(b1),F(b2)
•
1
1
2
E A12 [( F (a1 )  F (a2 )) ]  EB [( F (b1 )  F (b2 )) 2 ]
4
4
•
F(b1), F(b2) are close to {-1,1}
From Gap instances to Gap instances
Instance
SDP = c
OPT = s
A Gap Instance for
the Strong SDP for
CSP
Dictatorship
Test
Completeness = c
Soundness = s
UG
Hardness
Completeness = c
Soundness = s
UG Gap
instance for a
Strong SDP
2-CSP over {0,..q-1}
For each variable u in CSP,
Introduce q variables : {u0 , u1 ,.. uq-1 }
uc = 1,
u=c
ui = 0 for i≠c
Payoff for u,v :
P(u,v) = ∑a ∑b P(a,b)ua vb
2-CSP over {0,..q-1}
Total PayOff
Arbitrary k-ary GCSP
SDP is similar to the one obtained by k-rounds of
Lasserre
Rounding Scheme
(For Boolean CSPs)
Rounding Scheme was discovered by the
reversing the soundness analysis.
This fact was independently observed by Yi Wu
SDP Rounding Schemes
SDP Vectors
(v1 , v2 .. vn )
Random Projection
Projections
(y1 , y2 .. yn )
For any CSP, it is enough to
do the following:
Instead of one random
projection, pick sufficiently
many projections
Process the projections
Assignment
Use a multilinear
polynomial P to process the
projections
Rounding By Polynomial P(y1,… yR)
Roughly
Formally
Sample R Random Sample R independent vectors : w(1), w(2) ,.. w(R)
Directions
Each with i.i.d Gaussian components.
Project each vi along all directions w(1), w(2) ,..
Project along
w(R)
them
Yi(j) = v0∙vi + (1-ε)(vi – (v0∙vi)v0) ∙ w(j)
Compute P on
projections
Compute
xi = P(Yi(1) , Yi(2) ,.. Yi(R))
Round the output If xi > 1,
xi = 1
of P
If xi < -1,
xi = -1
If xi is in [-1,1]
xi = 1 with probability (1+xi)/2
-1 with probability (1-xi)/2
Algorithm
R is a constant parameter
Solve SDP(III) to obtain vectors (v1 ,v2 ,… vn )
Smoothen the SDP solution (v1 ,v2 ,… vn )

For all multlinear polynomials
P(y1 ,y2, .. yR) do
Round using P(y1 ,y2, .. yR)
Output the best solution obtained
Discretization
“For all multilinear polynomials P(y1 ,y2, .. yR)
do”
- All multilinear polynomials with coefficients
bounded within [-1,1]
- Discretize the set of all such multi-linear
polynomials
There are at most a constant number of such
polynomials.
Smoothening SDP Vectors
Let u1 ,u2 .. un denote the SDP vectors
corresponding to the following distribution
over integral solutions:
``Assign each variable uniformly and
independently at random”
Substitute
vi* ∙ vj* = (1-ε) (vi ∙ vj) + ε (ui∙ uj)
Non-Boolean CSPs
There will be q rounding polynomials instead of one polynomial.
Projection is in the same fashion:
Yi(j) = v0∙vi + (1-ε)(vi – (v0∙vi)v0) ∙ w(j)
To Round the Output of the polynomial, do the following:
From Gap instances to Gap instances
Instance
SDP = c
OPT = s
Worst Case
Instance
Dictatorship
Test
Completeness = c
Soundness = s
A Gap Instance for
the Strong SDP for
CSP
UG
Hardness
Completeness = c
Soundness = s
UG Gap
instance for a
Strong SDP
Backup Slides
Rounding for larger domains
Remarks
For every CSP and every ε > 0, there is a large
enough constant R such that
• Approximation achieved is within ε of optimal
for all CSPs if Unique Games Conjecture is
true.
• For 2-CSPs, the approximation ratio is within ε
of the integrality gap of the SDP(I).
Rounding Schemes
Very different rounding schemes for every CSP.
with often complex analysis.
Max Cut - Random hyperplane cutting
Multiway cut - Complicated Cutting the simplex.
• Our algorithm is a generic rounding procedure.
• Analysis does not compute the approximation
factor, but indirectly shows that it is equal to the
integrality gap.
“Sample R independent vectors : w1, w2 ,.. wR
each with i.i.d Gaussian components.
For all multlinear polynomials P(y1 ,y2, .. yR) do
Compute xi = P(vi∙ w1 , vi∙ w2 ,.. vi∙ wR)”
Goemans-Williamson rounding uses one single
random projection, this algorithm uses a
constant number of random projections.
Semidefinite Programming
• Linear program over the inner products
• Strongest algorithmic tool in approximation
algorithms
• Used in a large number of algorithms.
Integrality gap of a SDP relaxation =
Worst case ratio of
Integral Optimum
SDP Optimum
More Constraints?
Most SDP algorithms use simple relaxations
with few constraints.
[Arora-Rao-Vazirani] used the triangle
inequalities to get sqrt(log n) approximation
for sparsest cut.
Can the stronger SDPs yield better
approximation ratios for problems of interest?
Max Cut
Input : a weighted
graph G
10
15
7
1
1
3
Find a cut that
maximizes the number
of crossing edges
Max Cut SDP
Quadratic Program
Semidefinite
Program
-1
10
1
15
-1
1
7
1
1
1
3
-1
-1
-1
-1
Variables : vx1 , xv2 … xvn
x|i =vi1|2or= -1
1
Maximize
1
2
w
(
|
x
v

v
x
)
|

ij
ij
ii
j
4 (i , j )E
Relax all the xi to be unit vectors instead of {1,-1}.
All products are replaced by inner products of vectors
Max Cut SDP
Semidefinite Program
-1
10
1
15
-1
1
7
1
1
1
3
-1
-1
-1
-1
Variables : v1 , v2 … vn
| v i |2 = 1
Maximize
1
2
w
|
v

v
|

ij
i
j
4 (i , j )E