  We present a new greedy rounding algorithm for the Cycle Packing Problem for
uncrossable cycle families in planar graphs. This improves the best-known upper
bound for the integrality gap of the natural packing LP to a constant slightly
less than 3.5. Furthermore, the analysis works for both edge- and
vertex-disjoint packing. The previously best-known constants were 4 for
edge-disjoint and 5 for vertex-disjoint cycle packing. This result also
immediately yields an improved Erd\H{o}s-P\'osa ratio: for any uncrossable
cycle family in a planar graph, the minimum number of vertices (edges) needed
to hit all cycles in the family is less than 8.38 times the maximum number of
vertex-disjoint (edge-disjoint, respectively) cycles in the family.
  Some uncrossable cycle families of interest to which the result can be
applied are the family of all cycles in a directed or undirected graph, in
undirected graphs also the family of all odd cycles and the family of all
cycles containing exactly one edge from a specified set of demand edges. The
last example is an equivalent formulation of the fully planar Disjoint Paths
Problem. Here the Erd\H{o}s-P\'osa ratio translates to a ratio between integral
multi-commodity flows and minimum cuts.
