Motivated by mirror symmetry, many researchers have been studying the relation among the space of stability conditions, (the set of) semi-orthogonal decompositions and a certain Frobenius manifold. Macri and Dimitrov–Katzarkov investigated stability conditions arising from certain full exceptional collections to study the topological structure of the space. On the other hand, such full exceptional collections also play an important role in the representation theory of algebras. In this talk, I will explain their relation in the case of the derived category of an acyclic quiver. I also show that, for a given stability condition on the derived category of a finite acyclic quiver, there exists a compatible full exceptional collection (in the sense of Dimitrov–Katzarkov). This talk is based on joint work with Wu Dongjian.