SDP gaps and UGC-hardness for MAXCUTGAIN

Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson semidefinite programming (SDP) algorithm by M. Goemans and D. Williamson (1995) is guaranteed to find a cut of size .878 middot c. However this guarantee becomes trivial when c is near frac12, since a random cut has expected size frac12. Recently, M. Charikar and K. Worth (2004) (analyzing an algorithm of U. Feige and G. Langberg (2001)) showed that given a graph with maximum cut frac12 + epsiv, one can find a cut of size frac12 + Omega(epsiv/ log(1/epsiv)). The main contribution of our paper is twofold: 1. We give a natural frac12 + epsiv vs. frac12 + O(epsiv/ log(1/epsiv)) SDP gap for MAXCUT in Gaussian space. This shows that the SDP-rounding algorithm of Charikar-Worth is essentially best possible. Further, the "s-linear rounding functions" used in the works of M. Charikar and K. Worth (2004) and U. Freige and M. Langberg (2001) arise as optimizers in our analysis, somewhat confirming a suggestion of U. Freige and M. Langberg (2001). 2. We show how this SDP gap can be translated into a long code test with the same parameters. This implies that beating the Charikar-Worth guarantee with any efficient algorithm is NP-hard, assuming the unique games conjecture (UGC) by S. Khot (2002). We view this result as essentially settling the approximability of MAXCUT, assuming UGC. Building on (1) we show how "randomness reduction" on related SDP gaps for the QUADRATICPROGRAMMING programming problem lets us make the Omega(log(1/epsiv)) gap as large as Omega(log n) for n-vertex graphs. In addition to optimally answering an open question of N. Alen et al. (2006), this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our technique in (2), we also show how to translate Reeds's SDP gap by J. Reeds (1993) for the Grothendieck Inequality into a UGC-hardness result for computing the par middot parinfin rarr 1 norm of a matrix