  We study the cycle-valued reduced Gromov-Witten theory of a nonsingular
projective K3 surface. For primitive curve classes, we prove that the
correspondence induced by the reduced virtual fundamental class respects the
tautological rings. Our proof uses monodromy over the moduli space of K3
surfaces, degeneration formulae and virtual localization. As a consequence of
the monodromy argument, we verify an invariance property for Gromov-Witten
invariants of K3 surfaces in primitive curve class conjectured by
Oberdieck-Pandharipande.
