Generic Conversion of SDP gaps to Dictatorship Test (for Max Cut) Venkatesan Guruswami Fields Institute Summer School June 2011 (Slides borrowed from Prasad Raghavendra)

Dictatorship Test 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 a dictator function F(x 1,… x R ) = x i 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 Completeness Value of Dictator Cuts F(x) = x i Soundness The 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

v1v1 v2v2 v3v3 v4v4 v5v5 Recall Max Cut SDP: Embed the graph on the n-dimensional unit ball, Maximizing ¼ ( Average Squared Length of the edges )

Overview v1v1 v2v2 v3v3 v4v4 v5v R-dimensional hypercube R =large constant Graph G SDP Solution Completeness Value of Dictator Cuts = SDP Value (G) Soundness Given 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

Graph construction v1v1 v2v2 v3v3 v4v4 v5v5 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 Cuts F(x) = x i 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

Dictator Cuts R-dimensional hypercube v1v1 v2v2 v u v5v5 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 = X Y = Fraction of edges cut by dictator = ¼ Average Squared Distance

Sphere graph associated with G v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 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. v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 SDP Value (Graph G) = SDP Value ( Sphere Graph H)

v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 MaxCut (H) = S MaxCut (G) ≥ S Pick a random rotation of G and read the cut induced on it. Thus, v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 MaxCut (H) ≤ MaxCut(G)

Cuts far from Dictators v1v1 v2v2 v3v3 v4v4 v5v5 R-dimensional hypercube 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) v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5

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)