Inapproximability of MAX-CUT Khot,Kindler,Mossel and O ’ Donnell Moshe Ben Nehemia June 05

Main Result  It is NP-Hard problem to approximate MAX-CUT to within a factor  is the approximation ratio achieved by the algorithm of Goemans & Williamson.  The result follows from:  1. Unique Games conjecture  2. Majority is Stablest Theorem

Hardness of Approximation  History:  Bellare & Goldreich & Sudan :It is NP Hard to approximate MAX-CUT within factor higher than 83/84  Hasted improved the result to 16/17  Today: closing the gap …

Introduction  MAX-CUT: Given a Graph G =(V,E), find a partition C=(V 1, V 2 ) that maximize:  Unique Label Cover: Given a bi-partite graph with left side vertices- V,right side W, and edges- E each edge have a constraint bijection The goal: assign each vertex a label which satisfy the constraint.

Unique Games Conjecture: For any there exist a constant Such that it is NP-hard to distinguish whether the Unique Label Cover problem with label set in size M has optimum at least or at most

Some defintions  Let be an arbitrary boolean function  The influence of x i on f  Let x be a uniformly random string in :E[X]=0 and form y by flipping each bit with prob The noise stability of f for a noise rate is:

 The Correlation between x,y is define to be: E[XY] = 2 Pr[X=Y]-1  Let x be a uniformly random string in y be -correlated copy :i.e. pick each bit independently s.t. The noise correlation of f with parameter is:

 Result[60 ’ ] :

 Fix then for any there is a small enough s.t. if is any function satisfying : Then: The Majority is Stablest Theorem

On the Geometry of MAX-CUT  The Goemans-Williamson algorithm:  Embedding the graph in the unit sphere of R n :  The embedding is selected s.t. this sum is maximize A cut in G is obtained by choosing a random hyperplane through the origin. And this sum bounds from above the size of the maximal cut

On the Geometry of MAX-CUT The probability that vertics u,v lie on opposite sides of the cut is: So the expected weight is

On the Geometry of MAX-CUT  So to get:  Set the approximation ratio to:

Reminder  The Long Code:  The codeword encoding the message is by the truth table of the “ dictator ” function:

Technical Background  The Bonami Beckner operator Proposition: Let and then:

Technical Background Proposition: Let then for every Proof: Define: And : And using the Parseval identity we get the proposition

Technical Background  Let and let The k-degree influence of coordinate i on f is defined by: Proposition: The “ Majority is Stablest ” Theorem remains true if we change the assumption to

Reverse version of the “Majority is Stablest”  Fix then for any there is a small enough s.t. if is any function satisfying : Then:

Reverse version of the “Majority is Stablest”  Proof:  Take such f, and define:  Now g holds:  And now apply the original Theorem

Reduction from Unique LC to MAX-CUT Notations: denote the string and xy the coordinatewise product of x and y  Lemma 1: Completeness  If ULC have OPT then MAX-CUT have cut  Lemma 2: Soundness  If ULC have OPT then MAX-CUT have cut at most

Reduction from Unique LC to MAX-CUT Unique Label Cover WV v W’W’ w MAX-CUT j J’J’ i {-1,1} M

Reduction from Unique LC to MAX-CUT  The Reduction:  Pick a vertex at random and 2 of its neighbors:  Let and be the constrains for those edges  Let f,g be the supposed Long Codes of the labels  Pick at random  Pick by choosing each coordinate independently to be 1 with probability and -1 with prob. Edge in Cut iff

Reduction from Unique LC to MAX-CUT  Completeness Assume that the LC instance has a labeling which satisfies fraction of the edges. now encode these labels via Long Code with prob both the edges are satisfied by the labeling Denote the label of v,w,w ’ by i,j,j ’

Reduction from Unique LC to MAX-CUT  Completeness note that: Now f,g are the Long Codes of j,j ’, so: The two bits are unequal iff and that happens with prob. hence the completeness :

Reduction from Unique LC to MAX-CUT  Soundness – The Proof Strategy if the max-cut bigger than we ’ ll be able to “ decode ” the “ Long Code “ and create a labeling which satisfy significant fraction of the edges in the LC problem, and get a contradiction by choosing small enough.

Reduction from Unique LC to MAX-CUT From the Fourier Transform:

Reduction from Unique LC to MAX-CUT  The expectation over x vanishes unless and then s,s ’ have the same size. Because: We got: Because of for at least v in V ( “ good ” v) We have

Reduction from Unique LC to MAX-CUT Define Now:

Reduction from Unique LC to MAX-CUT  Now,from the “ Majority is stablest ” theorem:  We conclude that h has at least one coordinate j s.t. label the vertex v with j And we have:

Reduction from Unique LC to MAX-CUT  From the above equation we have that for at least fraction of neighbors w of v we have Define And so, Because we got that

Reduction from Unique LC to MAX-CUT  Now,if we label each vertex w in W by random element from Cand[w], then among the “ good ” vertices v at least satisfied. or among the edges, and that yields the contradiction

Restriction to the cube  For 2 vectors x,y on the cube,define the weight of (x,y) to be: Pr[X=x and Y=y] where X,Y are -correlated elements on the cube  A cut C defines a boolean function on the cube f c.  The size of C is exactly  The expected size of the maximal cut is  But, the “ dictator cut “ has size The “ Majority is stablest ” conjecture claim that these are the only cases.

The strategy of the proof  correctness A legal code word (dictator func) has Pr[acc]=  Soundness let be a func far from being a “ long code ” from the Majority is Stablest we get so the test pass with prob: