On a general framework for network representability in discrete optimization

TL;DR

This paper introduces a comprehensive framework for understanding which functions can be represented as network cut functions in discrete optimization, extending previous results to functions on arbitrary finite sets.

Contribution

It provides a complete characterization of network representable functions on binary domains and applies expressive power theory to analyze representability in this new framework.

Findings

Complete characterization of network representable functions on ,1^n.

Identification of certain ternary bisubmodular and binary k-submodular functions that are not network representable.

Extension of the theory of expressive power to the new framework for network representability.

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. \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.