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

The Lovaacutesz thetav-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} isin 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 thetav-function with an additive error of delta > 0, which runs in time O(alpha
²
n
²
/delta
²
log n middot M
_e
), where alpha = thetav(G) and M
_e
= O(n
³
) is the time for a matrix exponentiation operation. Moreover, our techniques generalize to the weighted Lovasz thetav-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+epsi) in time O(epsi
^-2
n
⁵
log n).