On the Lovász Theta function for Independent Sets in Sparse Graphs
Nikhil Bansal, Anupam Gupta, Guru Guruganesh · 2015
We consider the maximum independent set problem on graphs with maximum degree d. We show that the integrality gap of the Lovasz Theta function-based SDP has an integrality gap of O~(d/log3/2 d). This improves on the previous best result of O~(d/log d), and narrows the gap of this basic SDP to the integrality gap of O~(d/log2 d) recently shown for stronger SDPs, namely those obtained using poly log(d) levels of the SA+ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r.