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Prasad RaghavendraUniversity of Washington
Seattle
Optimal Algorithms and Inapproximability Results for
Every CSP?
Constraint Satisfaction ProblemA Classic Example : Max-3-SAT
Given a 3-SAT formula,Find an assignment to the variables that satisfies the maximum number of clauses.
))()()(( 145532532321 xxxxxxxxxxxx Equivalently the
largest fraction of clauses
Variables : {x1 , x2 , x3 ,x4 , x5} Constraints : 4 clauses
Constraint Satisfaction Problem
Instance :• Set of variables.• Predicates Pi applied on variables
Find an assignment that satisfies the largest fraction of constraints.
Problem :
Domain : {0,1,.. q-1}Predicates : {P1, P2 , P3 … Pr}
Pi : [q]k -> {0,1}
Max-3-SAT
Domain : {0,1}Predicates :
P1(x,y,z) = x ѵ y ѵ z
))()()(( 145532532321 xxxxxxxxxxxx
Generalized CSP (GCSP)
Replace Predicates by Payoff Functions (bounded real valued)
Problem :
Domain : {0,1,.. q-1}Pay Offs: {P1, P2 , P3 … Pr}
Pi : [q]k -> [-1, 1]Pay Off Functions can be Negative
Can model Minimization Problems like Multiway Cut, Min-Uncut.
Objective :
Find an assignment that maximizes the
Average Payoff
Examples of GCSPs
Max-3-SATMax CutMax Di CutMultiway CutMetric Labelling
0-ExtensionUnique Gamesd- to - 1 GamesLabel CoverHorn Sat
Unique GamesA Special Case
E2LIN mod pGiven a set of linear equations of the form:
Xi – Xj = cij mod p
Find a solution that satisfies the maximum number of equations.
x-y = 11 (mod 17)x-z = 13 (mod 17)
…….
z-w = 15(mod 17)
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.
Hardness Results: No constant factor approximation for unique games. [Feige-Reichman]
Algorithm On (1-Є) satisfiable instances
[Khot 02]
[Trevisan]
[Gupta-Talwar] 1 – O(ε logn)
[Charikar-Makarychev-Makarychev]
[Chlamtac-Makarychev-Makarychev]
[Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi]
)2/( p)loglog(1 pnO
)log(1 3 nO
))/1log((1 5/12 pO
1log1
Why is UGC important?Problem Best
Approximation Algorithm
NP Hardness Unique Games Hardness
Vertex CoverMax CUTMax 2- SATSPARSEST CUTMax k-CSP
20.878
0.9401
1.360.941
0.95461+ε
20.878
0.9401Every Constant
nlog
kk 2/ kkO 2/2 kkO 2/
UG hardness results are intimately connected to the limitations of Semidefinite Programming
Max Cut
10
15
3
7
11
Input : a weighted graph G
Find a cut that maximizes the number of crossing edges
Max Cut SDP
Quadratic Program
Variables : x1 , x2 … xn
xi = 1 or -1
Maximize
10
15
3
7
11
1
1
1
-1
-1
-1
-1-1
-1
Eji
jiij xxw),(
2)(4
1
Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors
Semidefinite Program
Variables : v1 , v2 … vn
| vi |2 = 1
Maximize
Eji
jiij vvw),(
2||4
1
MaxCut Rounding
v1
v2
v3
v4
v5
Cut the sphere by a random hyperplane, and output the induced graph cut.
- A 0.878 approximation for the problem.
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
Two CurvesIntegrality Gap CurveS(c) = smallest value of the integral solution, given SDP value c.
UGC Hardness CurveU(c) = The best polytime computable solution, assuming UGC given an instance with value c.
0 1Optimum Solution
S(c)
U(c)
Fix a GCSP
If UGC is true:U(c) ≥ S(c)
If UGC is false:U(c) is
meaningless!
UG Hardness Result
Roughly speaking,Assuming UGC, the SDP(I), SDP(II),SDP(III) give best
possible approximation for every CSP
c = SDP ValueS(c) = SDP Integrality GapU(c) = UGC Hardness Curve
Theorem 1:For every constant η > 0, and every GCSP Problem,
U(c) < S(c+ η) + η
0 1Optimum Solution
S(c)
U(c)
U(c)
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
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
c = SDP ValueS(c) = SDP Integrality GapU(c) = UGC Hardness Curve
0 1Optimum Solution
S(c)
U(c)
0 1Optimum Solution
S(c)U(c)
NP-hard
algorithm
If UGC is true, then for every Generalized Constraint Satisfaction Problem :
If UGC is false??
•Hardness result doesn’t make sense.
•How good is the rounding scheme?
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)
S(c) = SDP Integrality Gap
Theorem: Let A(c) be rounding scheme’s performance on input with SDP value = c. For every constant η > 0
A(c) > S(c- η) - η0 1Optimum Solution
S(c)
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 [Goemans-Williamson]Max-2-SAT [Lewin-Livnat-Zwick]Unique Games [Charikar-Makarychev-Makarychev]
Computing Integrality Gaps
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)
0 1Optimum Solution
S(c)
Explicit bounds on the size of an integrality gap instance for any 2-CSP.
Related WorkProblem Best
Approximation Algorithm
Unique Games Hardness
Vertex CoverMax CUTMax 2- SATSPARSEST CUTMax k-CSP
20.878
0.9401
2 [Khot-Regev] 0.878 [Khot-Kindler-Mossel-O’donnell]0.9401 [Per Austrin]Every Constant [Chawla-Krauthgamer-..] [Trevisan-Samorodnitsky]
kk 2/ kkO 2/nlog
[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
Dictatorship TestGiven a function F : {-1,1}R {-1,1}•Toss random coins•Make a few queries to F •Output either ACCEPT or REJECT
F is a dictator functionF(x1 ,… xR) = xi
F is far from every dictator function
(No influential coordinate)
Pr[ACCEPT ] = Completeness
Pr[ACCEPT ] =Soundness
ConnectionsSDP Gap Instance
SDP = 0.9OPT = 0.7
UG Hardness
0.9 vs 0.7
Dictatorship Test
Completeness = 0.9Soundness = 0.7
[Khot-Kindler-Mossel-O’Donnell]
[Khot-Vishnoi]For sparsest cut, max cut.[This Paper]
All these conversions hold for every GCSP
A Dictatorship Test for Maxcut
CompletenessValue of Dictator Cuts
F(x) = xi
SoundnessThe maximum value attained by a cut far from a dictator
A dictatorship test is a graph G on the hypercube.A cut gives a function F on the hypercube
Hypercube = {-1,1}100
An Example : Maxcutv1
v2
v3
v4
v5
10
15
3
71
1
100 dimensional hypercube
Graph G SDP Solution
CompletenessValue of Dictator Cuts =
SDP Value (G)
SoundnessGiven a cut far from every dictator :It gives a cut on graph G with the same value.
In other words, Soundness ≤ OPT(G)
From Graphs to Tests10
15
3
7
11
v1
v2
v3
v4
v5
Graph G (n vertices)
100 dimensional hypercube : {-1,1}100
SDP Solution
For each edge e, connect every pair of vertices in hypercube separated by the length of e
Constant independent of
size of G
H
Completeness
Echoice of edge e=(u,v) in G
[EX,Y in 100 dim hypercube with dist |u-v|^2 [ (F(X)-F(Y))2 ] ]
v1
v2
v3
v4
v5
100 dimensional hypercube
-1
-1-1
1
1
1
For each edge e, connect every pair of vertices in hypercube separated by the length of e
Set F(X) = X1
(X1 – Y1)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 variableshas similar distribution as
Sum of large number of Gaussian random variables.”
From Hypercube to the Sphere
100Dimensional hypercube
100 dimensio
nal sphere
F : [-1,1]
Express F as a multilinear polynomial using Fourier expansion, thus extending it to the sphere.
P : Real numbers
Since F is far from a dictator, by invariance principle, its behaviour on the sphere is similar to its behaviour on hypercube.
Nearly always [-1,1]
A Graph on the Sphere10
15
3
7
11
v1
v2
v3
v4
v5
Graph G (n vertices)
100 dimensional sphere
SDP Solution
For each edge e, connect every pair of vertices in sphere separated by the length of e
S
Hypercube vs Sphere
H S
F:{-1,1}100 -> {-1,1} is a cut far from every dictator.
P : sphere -> Nearly {-1,1}Is the multilinear extension of F
By Invariance Principle,
MaxCut value of F on H ≈ Maxcut value of P on S.
Soundnessv1
v2
v3
v4
v5 For each edge e in the graph G connect every pair of vertices in hypercube separated by the length of e
SG
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
Roughly FormallySample R Random Directions
Sample R independent vectors : g(1), g(2) ,.. g(100) Each with i.i.d Gaussian components.
Project along them
Project each vi along all directions g(1), g(2) ,.. g(100)
Yi(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) g∙ (100)
Compute P on projections
Compute xi = P(Yi
(1) , Yi(2) ,.. Yi
(100))Round the output of P
If xi > 1, xi = 1 If xi < -1, xi = -1 If xi is in [-1,1]
xi = 1 with probability (1+xi)/2 -1 with probability (1-xi)/2
Given the Polynomial P(y1,… y100)
Key Lemma
Any CSP Instance
G
DICTGDictatorship Test
on functionsF : {-1,1}n ->{-1,1}
If F is far from a dictator,RoundF (G) ≈ DICTG (F)
1) Tests of the verifier are same as the constraints in instance G2) Completeness = SDP(G)
Any Function
F: {-1,1}n → {-1,1}
RoundFRounding Schemeon CSP Instances G
UG Hardness Result
InstanceSDP = cOPT = s
Dictatorship Test
Completeness = cSoundness <= s
UG Hardness
Completeness = cSoundness <= s
Worst Case Gap Instance
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
Instance ISDP = cOPT = ?
AnyDictatorship
TestCompleteness = c
UG Hardness
Completeness = c
Soundness of any Dictatorship Test ≥ U(c)
There is some function F : {0,1}R -> {0,1} that hasPr[F is accepted] ≥ U(c)
By Key Lemma, Performance of F as rounding polynomial on instance I = Pr[F is accepted] > U(c)
Dictatorship Test (I)
Completeness = c
Related Developments
• Multiway Cut and Metric Labelling problems.
• Maximum Acyclic Subgraph problem
• Bipartite Quadratic Optimization Problem (Computing the Grothendieck constant)
[Manokaran, Naor, Schwartz, Raghavendra]
[Guruswami,Manokaran, Raghavendra]
[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.
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
v1
v2
v3
v4
v5
Cut the sphere by a random hyperplane, and output the induced graph cut.
Equivalently,
•Pick a random direction g.•For each vector vi , project vi along g
yi = vi . g•Assign
xi = 1 if yi > 0xi = 0 otherwise.
SDP Rounding Schemes
SDP Vectors (v1 , v2 .. vn )
Projections(y1 , y2 .. yn )
Assignment
Random Projection
Process the projections
For any CSP, it is enough to do the following:
Instead of one random projection, pick sufficiently many (say 100) projections
Use a multi linear polynomial P to process the projections
UG Hardness Results
InstanceSDP = cOPT = s
Dictatorship Test
Completeness = cSoundness <= s
UG Hardness
Completeness = cSoundness <= s
Worst Case Gap Instance
Theorem 1:For every constant η > 0, and every GCSP Problem,
U(c) < S(c+ η) + η
Multiway Cut and Labelling Problems
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.
10
15
3
7
11
3-Way Cut:Separate the 3-terminals while
separating the minimum number of edges
[Manokaran, Naor, Schwartz, Raghavendra]
Maximum Acyclic Subgraph
Given a directed graph, order the vertices to maximize the number of forward edges.
[Guruswami,Manokaran, Raghavendra]
Theorem: Assuming Unique Games Conjecture,The best algorithm’s output is as good as a random ordering.
Theorem: Unconditionally, the simple SDP does not give better approximations than random.
The Grothendieck Constant
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.
[Raghavendra,Steurer]
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
InstanceSDP = cOPT = s
Dictatorship Test
Completeness = cSoundness <= s
UG Hardness
Completeness = cSoundness <= s
Worst Case Gap Instance
Best UG Hardness =
Integrality GapU(c) < S(c+η) + η
Algorithm
Instance ISDP = cOPT = ?
AnyDictatorship
TestCompleteness = c
UG Hardness
Completeness = c
Soundness of any Dictatorship Test ≥ U(c)
There is some function F : {0,1}R -> {0,1} that hasPr[F is accepted] ≥ U(c)
By Key Lemma, Performance of F as rounding polynomial on instance I = Pr[F is accepted] > U(c)
Dictatorship Test (I)
Completeness = c
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
Unconditional Results For 2-CSPs
Dictatorship Test(I)
Completeness = cSoundness = s
UG Hardness
Completeness = cSoundness = s
UG Integrality Gap instance
Integrality Gap instanceSDP = cOPT ≤ sAlgorithm’s performance
matches the integrality gap of the SDP
[Khot-Vishnoi]
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
CompletenessThis 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
1
2132
322
21 ||||||3
1vvvvvv
SDP:Variables : v1 , v2 ,v3
|v1|2 = |v2|2 = |v3|2 =1
Maximize2 3
E[a1 a2] = v1 v∙ 2
E[a12] = |v1|2 E[a2
2] = |v2|2
For every edge, there is a local distribution over integral solutions such that:All the moments of order at most 2 match the inner products.
Local Random Variables
1
32
Fix an edge e = (1,2).
There exists random variables a1 a2 taking values {-1,1} such that:
c = SDP Valuev1 , v2 , v3 = SDP Vectors
A12A13
A23
Dictatorship TestPick 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 Valuev1 , v2 , v3 = SDP Vectors
A12,A23,A31 = Local Distributions
1
32
A12
Input Function:F : {-1,1}R -> {-1,1}
Max Cut Instance
AnalysisPick 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
1
32
Max Cut Instance
]))()([(
4
1]))()([(
4
1]))()([(
4
1
3
1 213
232
221 312312
aFaFEaFaFEaFaFE AAA
Input Function:F : {-1,1}R -> {-1,1}
]))()([(
4
1]))()([(
4
1]))()([(
4
1
3
1 213
232
221 312312
aFaFEaFaFEaFaFE AAA
CompletenessA12,A23,A31 = Local Distributions
Input Function is a Dictator : F(x) = x1
])[(
4
1])[(
4
1])[(
4
1
3
1 21131
23121
22111 312312
aaEaaEaaE AAA
Suppose (a1 ,a2) is sampled from A12 then :E[a11 a21] = v1 v∙ 2 E[a11
2] = |v1|2 E[a212] = |v2|2
221
221 ||])[(
12vvaaEA
Summing up, Pr[Accept] = SDP Value(v1 , v2 ,v3)
E[b1 b2] = v1 v∙ 2 E[b2 b3] = v2 v∙ 3 E[b3 b1] = v3 v∙ 1
E[b1
2] = |v1|2 E[b22] = |v2|2 E[b3
2] = |v3|2
There is a global distribution B=(b1 ,b2 ,b3) over real numbers such that:All the moments of order at most 2 match the inner products.
Global Random Variablesc = SDP Value
v1 , v2 , v3 = SDP Vectors
g = random Gaussian vector.(each coordinate generated by i.i.d normal variable)
b1 = v1 g∙b2 = v2 g∙b3 = v3 g∙
1
32
B
Rounding with Polynomials
Input Polynomial : F(x1 ,x2 ,.. xR)
Generate b1 = (b11 ,b12 ,… b1R)
b2 = (b21 ,b22 ,… b2R)
b3 = (b31 ,b32 ,… b3R)
with each coordinate (b1t ,b2t ,b3t) according to global distribution B
Compute F(b1),F(b2) ,F(b3)
Round F(b1),F(b2),F(b3) to {-1,1}
Output the rounded solution.
1
32
B
]))()([(
4
1]))()([(
4
1]))()([(
4
1
3
1 213
232
221 bFbFEbFbFEbFbFE BBB
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)
•
• F(b1), F(b2) are close to {-1,1}
]))()([(4
1]))()([(
4
1 221
22112
bFbFEaFaFE BA
From Gap instances to Gap instances
InstanceSDP = cOPT = s
Dictatorship Test
Completeness = cSoundness = s
UG Hardness
Completeness = cSoundness = s
UG Gap instance for a
Strong SDP
A Gap Instance for the Strong SDP for
CSP
For each variable u in CSP,Introduce q variables : {u0 , u1 ,.. uq-1 }
uc = 1,
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}
u = c
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 )
Projections(y1 , y2 .. yn )
Assignment
Random Projection
Process the projections
For any CSP, it is enough to do the following:
Instead of one random projection, pick sufficiently many projections
Use a multilinear polynomial P to process the projections
Roughly FormallySample R Random Directions
Sample R independent vectors : w(1), w(2) ,.. w(R) Each with i.i.d Gaussian components.
Project along them
Project each vi along all directions w(1), w(2) ,.. w(R)
Yi(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) w∙ (j)
Compute P on projections
Compute xi = P(Yi
(1) , Yi(2) ,.. Yi
(R))Round the output of P
If xi > 1, xi = 1 If xi < -1, xi = -1 If xi is in [-1,1]
xi = 1 with probability (1+xi)/2 -1 with probability (1-xi)/2
Rounding By Polynomial P(y1,… yR)
Algorithm
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
R is a constant parameter
“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.
Discretization
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
* v∙ j* = (1-ε) (vi v∙ j) + ε (ui u∙ j)
Non-Boolean CSPs
There will be q rounding polynomials instead of one polynomial.
Projection is in the same fashion: Yi
(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) w∙ (j)
To Round the Output of the polynomial, do the following:
From Gap instances to Gap instances
InstanceSDP = cOPT = s
Dictatorship Test
Completeness = cSoundness = s
UG Hardness
Completeness = cSoundness = s
UG Gap instance for a
Strong SDP
A Gap Instance for the Strong SDP for
CSPWorst Case
Instance
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 w∙ 1 , vi w∙ 2 ,.. vi w∙ R)”
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
10
15
3
7
11
Input : a weighted graph G
Find a cut that maximizes the number of crossing edges
Max Cut SDP
Quadratic Program
Variables : x1 , x2 … xn
xi = 1 or -1
Maximize
10
15
3
7
11
1
1
1
-1
-1
-1
-1-1
-1
Eji
jiij xxw),(
2)(4
1
Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors
Semidefinite Program
Variables : v1 , v2 … vn
| vi |2 = 1
Maximize
Eji
jiij vvw),(
2||4
1
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