  Erd\H{o}s and P\'{o}sa proved in 1965 that there is a duality between the
maximum size of a packing of cycles and the minimum size of a vertex set
hitting all cycles. Such a duality does not hold if we restrict to odd cycles.
However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to
half-integral packing. We prove a far-reaching generalisation of the theorem of
Reed; if the edges of a graph are labelled by finitely many abelian groups,
then there is a duality between the maximum size of a half-integral packing of
cycles whose values avoid a fixed finite set for each abelian group and the
minimum size of a vertex set hitting all such cycles.
  A multitude of natural properties of cycles can be encoded in this setting,
for example cycles of length at least $\ell$, cycles of length $p$ modulo $q$,
cycles intersecting a prescribed set of vertices at least $t$ times, and cycles
contained in given $\mathbb{Z}_2$-homology classes in a graph embedded on a
fixed surface. Our main result allows us to prove a duality theorem for cycles
satisfying a fixed set of finitely many such properties.
