Fourier analysis and inapproximability for MAX-CUT: a case study

Many statements in the study of computational complexity can be cast as statements about Boolean functions f : {−1, 1} → {−1, 1}. However, it was only very late in the last century that the analytic properties of such functions, often expressed via the Fourier transform on the Boolean hypercube, became the key ingredient in proofs of hardness of approximation. In this short survey, we give a brief overview of the history of this relationship between harmonic analysis and inapproximability for CSPs by zooming in on the particularly illustrative example of MAX-CUT. We summarize Hastad’s seminal ideas from [4], proving unconditional NP-hardness of ( + )-approximating MAX-CUT. 17 Then we take a detailed look at how Khot, Kindler, Mossel and O’Donnell [7] pushed these Fourier-analytic methods further to prove UGC-hardness of approximating MAXCUT to within any constant factor larger than αGW ≈ 0.878, the factor achieved by the famous Goemans-Williamson approximation algorithm. In particular, we’ll discuss the Majority is Stablest Theorem and the role it plays in their analysis, with the hope of making this connection between a purely analytic invariance principle and the surprising UGC-optimality of Goemans and Williamson’s algorithm – as well as the need for analytic methods in proving computational hardness – appear a little less mysterious than it might at first sight.