Max-Cut via Kuramoto-type Oscillators

02/09/2021

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by   Stefan Steinerberger, et al.

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We consider the Max-Cut problem. Let G = (V,E) be a graph with adjacency matrix (a_ij)_i,j=1^n. Burer, Monteiro Zhang proposed to find, for n angles {θ_1, θ_2, …, θ_n}⊂ [0, 2π], minima of the energy f(θ_1, …, θ_n) = ∑_i,j=1^n a_ijcos(θ_i - θ_j) because configurations achieving a global minimum leads to a partition of size 0.878 Max-Cut(G). This approach is known to be computationally viable and leads to very good results in practice. We prove that by replacing cos(θ_i - θ_j) with an explicit function g_ε(θ_i - θ_j) global minima of this new functional lead to a (1-ε)Max-Cut(G). This suggests some interesting algorithms that perform well. It also shows that the problem of finding approximate global minima of energy functionals of this type is NP-hard in general.