  We consider node-weighted survivable network design (SNDP) in planar graphs
and minor-closed families of graphs. The input consists of a node-weighted
undirected graph $G=(V,E)$ and integer connectivity requirements $r(uv)$ for
each unordered pair of nodes $uv$. The goal is to find a minimum weighted
subgraph $H$ of $G$ such that $H$ contains $r(uv)$ disjoint paths between $u$
and $v$ for each node pair $uv$. Three versions of the problem are
edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP) and
vertex-connectivity SNDP (VC-SNDP) depending on whether the paths are required
to be edge, element or vertex disjoint respectively. Our main result is an
$O(k)$-approximation algorithm for EC-SNDP and Elem-SNDP when the input graph
is planar or more generally if it belongs to a proper minor-closed family of
graphs; here $k=\max_{uv} r(uv)$ is the maximum connectivity requirement. This
improves upon the $O(k \log n)$-approximation known for node-weighted EC-SNDP
and Elem-SNDP in general graphs [Nutov, TALG'12]. We also obtain an $O(1)$
approximation for node-weighted VC-SNDP when the connectivity requirements are
in $\{0,1,2\}$; for higher connectivity our result for Elem-SNDP can be used in
a black-box fashion to obtain a logarithmic factor improvement over currently
known general graph results. Our results are inspired by, and generalize, the
work of [Demaine, Hajiaghayi and Klein, TALG'14] who obtained constant factor
approximations for node-weighted Steiner tree and Steiner forest problems in
planar graphs and proper minor-closed families of graphs via a primal-dual
algorithm.
