  We prove the integrality and finiteness of open BPS invariants of toric
Calabi-Yau 3-folds relative to Aganagic-Vafa outer branes, defined from open
Gromov-Witten invariants by the Labastida-Mari\~no-Ooguri-Vafa formula.
Specializing to disk invariants, we extend the open/closed correspondence of
Gromov-Witten invariants to BPS invariants and prove the integrality of a class
of genus-zero BPS invariants of toric Calabi-Yau 4-folds, thereby providing
additional examples for the conjecture of Klemm-Pandharipande.
