We show that curve enumeration invariants of complex threefolds with nef anti-canonical bundle are determined by their values on local curves. This statement and its proof are inspired by the proof of the Gopakumar-Vafa integrality conjecture by Ionel and Parker. The conjecture of Maulik, Nekrasov, Okounkov, and Pandharipande relating Gromov-Witten and Donaldson-Pandharipande-Thomas invariants is known for local curves by work of Bryan, Okounkov, and Pandharipande, hence holds for all complex threefolds with nef anti-canonical bundle (in particular, all Calabi-Yau threefolds).