  Given two points in the plane, a set of obstacles defined by closed curves,
and an integer $k$, does there exist a path between the two designated points
intersecting at most $k$ of the obstacles? This is a fundamental and
well-studied problem arising naturally in computational geometry, graph theory,
wireless computing, and motion planning. It remains $\textsf{NP}$-hard even
when the obstacles are very simple geometric shapes (e.g., unit-length line
segments). In this paper, we show that the problem is fixed-parameter tractable
($\textsf{FPT}$) parameterized by $k$, by giving an algorithm with running time
$k^{O(k^3)}n^{O(1)}$. Here $n$ is the number connected areas in the plane
drawing of all the obstacles.
