One of the most important properties of the Lovász theta function is that for any $\epsilon >0$, it can be approximated to within an $\epsilon$ additive error in polynomial time (by the Ellipsoid Method).

Moreover, for every graph $G$, it is well-known that

$$\alpha(G) \leq \vartheta(G) \leq \overline{\chi}(G),$$

where $\alpha(G)$ is the size of a largest independent in $G$, $\vartheta(G)$ is the Lovasz theta number of $G$, and $\overline{\chi}(G)$ is the chromatic number of the complement of $G$.

For perfect graphs, we have $\alpha(G) = \vartheta(G) = \overline{\chi}(G)$. In particular $\vartheta(G)$ is always an integer. Therefore, rounding an approximation of the Lovász theta function yields a polynomial-time algorithm to compute the chromatic number of perfect graphs. As far as I know, no other efficient algorithm is known.