I will pre­sent the al­go­rithm I am cur­rently work­ing on that uses ran­dom sam­pling and a com­bi­na­tion of poly­gon de­com­po­si­tion and $k$-lev­els to get a 2-ap­prox­i­ma­tion al­go­rithm for the min en­clos­ing ge­o­desic disc of $k$ points in sim­ple poly­gons. The poly­gon has $m$ ver­tices total, $r$ re­flex ver­tices, and the set of points $S$ that our $k$ points is to be cho­sen from has size $n$. After $O(m+(n + r)\log(r))$ pre­pro­cess­ing, the run­time of the cur­rent ap­proach is $O((n^{11/7}\log^2(n)+nr)\log(r))$.