Lecture 24 : Hardness of approximation of MAX - CUT April 10 , 2008

• The 1 vs 1 decision problem is easy for MAX-CUTas the problem is equivalent to testing if the graph is bipartite. • Goemans-Williamson's SDP based algorithm for MAX-CUTyields a 1 2 (1−ρ) vs (arccos(ρ))/π approximation for all ρ < −0.69. Replacing ρ with 2δ − 1 shows that the 1 − δ vs 1 − Θ(√ δ) problem is easy. If we think about MAX-CUTas MIN-UNCUT, this would mean that the δ vs Θ(√ δ) problem is easy. From here on, we prefer to view MAX-CUTas MIN-UNCUT. • Garg, Vazirani and Yannakakis[3] give an O(log n) factor approximation for the multi-cut problem which can be used to give and O(log n) factor approximation for the MIN-UNCUTproblem, as any α factor approximation for the multi-cut problem translates in to an α factor approximation for the MIN-UNCUTproblem. • Charikar et. al. [1] give a O(√ log n) factor approximation for MIN-UNCUT. We will now describe a 2-query long code test for testing dictator functions and use it in conjunction with the Unique Games Conjecture and the " Majority is Stablest " theorem to prove 1 2 − 1 2 ρ − η vs arccos(ρ) π + η hardness for the MAX-CUTproblem. 2 2-query long code test and the MIS theorem In order to test if the variables encoding the label for a vertex encode a valid label, we describe a 2-query long-code test for dictator functions. The test is as follows: • pick x ∈ {0, 1} k randomly. • Construct y by copying x in to y and flipping each bit in y independently with probability (1 − ρ)/2 (x and y are ρ-correlated). • Check if f (x)f (y) = −1.