On a General Framework for Network Representability in Discrete Optimization

Abstract

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. Živný-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.