Generic Conversion of SDP gaps to Dictatorship Test (for Max Cut)

Venkatesan GuruswamiFields Institute Summer School

June 2011

(Slides borrowed from Prasad Raghavendra)

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

UG Hardness

[Khot-Kindler-Mossel-O’Donnell]

A dictatorship test where • Completeness = and Soundness = α• the verifier’s tests are predicates from a CSP

It is UG-hard to (α+, -) –distinguish CSP

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}R

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Recall Max Cut SDP:

Embed the graph on the n-dimensional unit ball,

Maximizing

¼ (Average Squared Length

of the edges)

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R-dimensional hypercubeR =large constant

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 (nearly) the same value.

So soundness Max Cut (G)

Gap of test = integrality gap of SDP

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SDP Solution

R-dimensional hypercube : {-1,1}R

For each edge e, connect every pair of vertices in hypercube separated by the length of e

Formally, generate edges of expected squared length = d :

1) Starting with a random x Є {-1,1}R ,1) Generate y by flipping each bit of x with probability d/4

Output (x,y)

Dichotomy of Cuts

Dictator CutsF(x) = xi

Cuts Far From Dictators(influence of each coordinate on function F is small)

A cut gives a function F on the hypercube

F : {-1,1}R-> {-1,1}

Hypercube = {-1,1}R

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Dictator Cuts

R-dimensional hypercube

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For each edge e = (u,v), connect every pair of vertices in hypercube separated by the length of e

Value of Dictator Cuts = SDP Value (G)

Pick an edge e = (u,v), consider all edges in hypercube corresponding to e

Fraction of red edges cut by horizontal dictator .

Fraction of dictators that cut one such edge (X,Y)

Number of bits in which X,Y differ

=|u-v|2/4

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Fraction of edges cut by dictator = ¼ Average Squared Distance

Sphere graph associated with G

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SDP Value = Average Squared Length of an Edge

Transformations• Rotation does not change the SDP value.• Union of two rotations has the same SDP value

Sphere Graph H :Union of all possible rotations of G.

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SDP Value (Graph G) = SDP Value ( Sphere Graph H)

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MaxCut (H) = S

MaxCut (G) ≥ S

Pick a random rotation of G and read the cut induced on it.Thus,

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MaxCut (H) ≤ MaxCut(G)

Cuts far from Dictatorsv1

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R-dimensional hypercube

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Intuition:

Sphere graph : Uniform on all directions

Hypercube graph : Axis are special directions

If a cut does not respect the axis, then it should not distinguish between Sphere and Hypercube graphs (formalized by invariance principle)

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Hypercube vs Sphere

H

F:{-1,1}R -> {-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 hypercube ≈ Maxcut value of P on Sphere graph H

At most Max Cut(G)