On a general framework for network representability in discrete optimization

In discrete optimization, representing an objective function as an $s$-$t$ cut function of a network is a basic technique to design an efficient minimization algorithm. A network representable function can be minimized by computing a minimum $s$-$t$ cut of a directed network, which is a very easy and fastly solved problem. Hence it is natural to ask what functions are network representable. In the case of pseudo Boolean functions (functions on $\{0,1\}^n$), it is known that any submodular function on $\{0,1\}^3$ is network representable. \v{Z}ivn\'y--Cohen--Jeavons showed by using the theory of expressive power that a certain submodular function on $\{0,1\}^4$ is not network representable. In this paper, we introduce a general framework for the network representability of functions on $D^n$, where $D$ is an arbitrary finite set. We completely characterize network representable functions on $\{0,1\}^n$ in our new definition. We can apply the expressive power theory to the network representability in the proposed definition. We prove that some ternary bisubmodular function and some binary $k$-submodular function are not network representable.