An SDP primal-dual algorithm for approximating the Lovász-theta function

The Lovasz ϑ-function [Lov79] on a graph G = (V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and X ij = 0 whenever {i, j} ∈ E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale's primal-dual method for SDP to design an approximate algorithm for the ϑ-function with an additive error of δ ≫ 0, which runs in time O(α2n2/δ2 log n · M e ), where α = ϑ(G) and M e = O(n3) is the time for a matrix exponentiation operation. Moreover, our techniques generalize to the weighted Lovasz ϑ-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+∊) in time O(∊−2n5 log n).