Hermitian Laplacians and a Cheeger inequality for the Max-2-Lin problem

Abstract

We study spectral approaches for the Max-2-Lin(k) problem, in which we are given a system of m linear equations of the form x_i - x_j = c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations. We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a Max-2-Lin(k) instance I. We develop an \tilde{O}(kn^2) time algorithm that, for any (1-\eps)-satisfiable instance, produces an assignment satisfying a (1 - O(k)\sqrt{\eps})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)\eps)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.