  We study the interplay of the moduli of curves and the moduli of K3 surfaces
via the virtual class of the moduli spaces of stable maps. Using Getzler's
relation in genus 1, we construct a universal decomposition of the diagonal in
Chow in the third fiber product of the universal K3 surface. The decomposition
has terms supported on Noether-Lefschetz loci which are not visible in the
Beauville-Voisin decomposition for a fixed K3 surface. As a result of our
universal decomposition, we prove the conjecture of Marian-Oprea-Pandharipande:
the full tautological ring of the moduli space of K3 surfaces is generated in
Chow by the classes of the Noether-Lefschetz loci. Explicit boundary relations
are constructed for all kappa classes.
  More generally, we propose a connection between relations in the tautological
ring of the moduli spaces of curves and relations in the tautological ring of
the moduli space of K3 surfaces. The WDVV relation in genus 0 is used in our
proof of the MOP conjecture.
