Integrality of relative BPS state counts of toric Del Pezzo surfaces

Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured by the authors to be integers. For toric Del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng; and more generally for toric Calabi-Yau threefolds by Konishi.