We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having 𝑛n edges. We also prove that in polygons with holes, there is a constant 𝑐>0c>0 such that no polynomial-time algorithm can solve the problem within an approximation factor of 𝑐log𝑛clogn, unless P=NP. For the second problem, we present a (𝑘+ℎ)(k+h)-FPT algorithm for computing a shortest tour that sees 𝑘k specified points in a polygon with ℎh holes. We also present a 𝑘k-FPT approximation algorithm for this problem having approximation factor 2‾√2. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of 𝑐log𝑛clogn, for some constant 𝑐>0c>0, unless P=NP.