  A classical branch of graph algorithms is graph transversals, where one seeks
a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects
all copies of subgraphs~$F$ from a fixed family $\mathcal F$. Many such graph
transversal problems have been shown to admit polynomial-time approximation
schemes (PTAS) for planar input graphs $G$, using a variety of techniques like
the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al.,
SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These
techniques do not seem to apply to graph transversals with parity constraints,
which have recently received significant attention, but for which no PTASs are
known. In the even-cycle transversal (\ECT) problem, the goal is to find a
minimum-weight hitting set for the set of even cycles in an undirected graph.
  For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the
standard covering LP relaxation is $\Theta(\log n)$, and that adding sparsity
inequalities reduces the integrality gap to~10. Our main result is a
primal-dual algorithm that yields a $47/7\approx6.71$-approximation for ECT on
node-weighted planar graphs, and an integrality gap of the same value for the
standard LP relaxation on node-weighted planar graphs.
