  We consider two optimization problems in planar graphs. In Maximum Weight
Independent Set of Objects we are given a graph $G$ and a family $\mathcal{D}$
of objects, each being a connected subgraph of $G$ with a prescribed weight,
and the task is to find a maximum-weight subfamily of $\mathcal{D}$ consisting
of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given
an edge-weighted graph $G$, two sets $\mathcal{D},\mathcal{C}$ of vertices of
$G$, where vertices of $\mathcal{D}$ have prescribed weights, and a nonnegative
radius $r$. The task is to find a minimum-weight subset of $\mathcal{D}$ such
that every vertex of $\mathcal{C}$ is at distance at most $r$ from some
selected vertex. Via simple reductions, these two problems generalize a number
of geometric optimization tasks, notably Maximum Weight Independent Set for
polygons in the plane and Weighted Geometric Set Cover for unit disks and unit
squares. We present quasi-polynomial time approximation schemes (QPTASs) for
both of the above problems in planar graphs: given an accuracy parameter
$\epsilon>0$ we can compute a solution whose weight is within multiplicative
factor of $(1+\epsilon)$ from the optimum in time
$2^{\mathrm{poly}(1/\epsilon,\log |\mathcal{D}|)}\cdot n^{\mathcal{O}(1)}$,
where $n$ is the number of vertices of the input graph. Our main technical
contribution is to transfer the techniques used for recursive approximation
schemes for geometric problems due to Adamaszek, Har-Peled, and Wiese to the
setting of planar graphs. In particular, this yields a purely combinatorial
viewpoint on these methods.
