  This paper investigates the combinatorial $\alpha$-curvature for vertex
scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a
parameterized generalization of the classical combinatorial curvature. A
discrete uniformization theorem for combinatorial $\alpha$-curvature is
established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization
theorem for classical combinatorial curvature. We further introduce
combinatorial $\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow for
vertex scaling to find piecewise hyperbolic metrics with prescribed
combinatorial $\alpha$-curvatures. To handle the potential singularities along
the combinatorial curvature flows, we do surgery along the flows by edge
flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu,
we prove the longtime existence and convergence of combinatorial
$\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow with surgery, which
provide effective algorithms for finding piecewise hyperbolic metrics with
prescribed combinatorial $\alpha$-curvatures.
