We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question
for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform
technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all
tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting
tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.