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

T.-H. Hubert Chan, Kevin L. Chang, Rajiv Raman

Abstract: 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</sub> = 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /delta <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log n middot M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> ), where alpha = thetav(G) and M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> = O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> log n).