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 [31]. 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 [13], 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.
Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time
