The Minimal model program (MMP, for short), which is a higher-dimensional analog of the classification method of surfaces, is a tool to find a ``simplest'' variety in each birational equivalence class. The MMP is also studied for schemes not necessarily defined over a field. Such a generalization of the MMP plays an important role to construct a nice model over an integer ring of a given variety. The MMP is known to hold for strictly semi-stable schemes over an excellent Dedekind scheme of relative dimension two whose each residue characteristic is neither 2 nor 3 by Kawamata. In this talk, I will introduce a generalization of the result of Kawamata without any assumption on the residue characteristics. This is based on a joint work with Teppei Takamatsu.