An optimal sdp algorithm for max-cut, and equally optimal long code tests

Let G be an undirected graph for which the standard Max-Cut SDP relaxation achieves at least a c fraction of the total edge weight, 1/2 ≤ c ≤ 1. If the actual optimal cut for G is at most an s fraction of the total edge weight, we say that (c, s) is an SDP gap. We define the SDP gap curve GapSDP: 1/2, 1 -> 1/2, 1 by GapSDP (c) = infs: (c, s) is an SDP gap. In this paper we complete a long line of work 15, 14, 20, 36, 19, 17, 13, 28 by determining the entire SDP gap curve; we show GapSDP (c) = S (c) for a certain explicit (but complicated to state) function S. In particular, our lower bound GapSDP (c) - S (c) is proved via a polynomial-time - RPR2' algorithm. Thus we have given an efficient, optimal SDP-rounding algorithm for Max-Cut. The fact that it is RPR2 confirms a conjecture of Feige and Langberg 17. We also describe and analyze the tight connection between SDP gaps and Long Code tests (and the constructions of 25, 3, 4). Using this connection, we give optimal Long Code tests for Max-Cut. Combining these with results implicit in 27, 29 and ideas from 19, we derive the following conclusions: - The Max-Cut SDP gap curve subject to triangle inequalities is also given by S (c). - No RPR2 algorithm can be guaranteed to find cuts of value larger than S (c) in graphs where the optimal cut is c. (Contrast this with the fact that in the graphs exhibiting the c vs. S (c) SDP gap, our RPR2 algorithm actually finds the optimal cut. ) - Further, no polynomial-time algorithm of any kind can have such a guarantee, assuming P ≠ NP and the Unique Games Conjecture.