Inapproximability of MAX-CUT

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

2. 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

3. 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…

4. Introduction • MAX-CUT: Given a Graph G =(V,E), find a partition C=(V1,V2) 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.

5. 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

6. Some defintions • Let be an arbitrary boolean function • The influence of xi 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:

7. 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:

8. Result[60’] :

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

10. On the Geometry of MAX-CUT • The Goemans-Williamson algorithm: • Embedding the graph in the unit sphere of Rn : • 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

11. 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

12. On the Geometry of MAX-CUT • So to get: • Set the approximation ratio to:

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

14. Technical Background • The Bonami Beckner operator Proposition: Let and then:

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

16. 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

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

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

19. 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

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

21. 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

22. 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’

23. 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 :

24. 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.

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

26. 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

27. Reduction from Unique LC to MAX-CUT Define Now:

28. 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:

29. 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

30. 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