A Semidefinite Approach to the $K_i$ Cover Problem

We apply theta body relaxations to the $K_i$-cover problem and show polynomial time solvability for certain classes of graphs. In particular, we give an effective relaxation where all $K_i$-$p$-hole facets are valid, and study its relation to an open question of Conforti et al. For the triangle free problem, we show for $K_n$ that the theta body relaxations do not converge by $(n-2)/4$ steps; we also prove for all $G$ an integrality gap of 2 for the second theta body.