Lower bounds for Seshadri constants on blow ups of ${\mathbb{P}}^2$

Let $\pi :X_r\to {\mathbb{P}}^2$ be a blow up of ${\mathbb{P}}^2$ at $r$ distinct points $p_1,p_2,\dots ,p_r$. We study lower bounds for Seshadri constants of ample line bundles on $X_r$. First, we consider the case when the points lie on a curve of degree $d\le 3$, and the case when $r\le 8$. We then assume that the points are very general and show that $\epsilon (X_r)\ge \frac{1}{2}$ if the Strong SHGH conjecture is true.