Hermitian Laplacians and a Cheeger inequality for the Max-2-Lin problem
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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 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 O(kn^2) time algorithm that, for any (1-ε)-satisfiable instance, produces an assignment satisfying a (1 - O(k)√(ε))-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)ε)-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.
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