  Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a
straightforward discretization of smooth conformal structures on surfaces.
Combinatorial $\alpha$-curvature for vertex scaling of piecewise linear metrics
on surfaces is a discretization of Gaussian curvature on surfaces. In this
paper, we investigate the prescribing combinatorial $\alpha$-curvature problem
on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory for
piecewise linear metrics on surfaces and variational principles with
constraints, we prove some Kazdan-Warner type theorems for prescribing
combinatorial $\alpha$-curvature problem, which generalize the results obtained
by Gu-Luo-Sun-Wu on prescribing combinatorial curvatures on surfaces.
Gu-Luo-Sun-Wu conjectured that one can prove Kazdan-Warner's theorems via
approximating smooth surfaces by polyhedral surfaces. This paper takes the
first step in this direction.
