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-Wus 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.
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