A note on the Minimum Norm Point algorithm

TL;DR

This paper introduces a more efficient practical implementation of the Minimum Norm Point algorithm for submodular function minimization, significantly improving performance in practice despite lacking new worst-case theoretical bounds.

Contribution

It provides an improved, empirically faster implementation of the algorithm, enhancing practical efficiency for submodular minimization problems.

Findings

Performs an order of magnitude faster for certain functions

No known worst-case bounds, but high practical performance

Applicable to important functions like graph cut minimization

Abstract

We present a provably more efficient implementation of the Minimum Norm Point Algorithm conceived by Fujishige than the one presented in \cite{FUJI06}. The algorithm solves the minimization problem for a class of functions known as submodular. Many important functions, such as minimum cut in the graph, have the so called submodular property \cite{FUJI82}. It is known that the problem can also be efficiently solved in strongly polynomial time \cite{IWAT01}, however known theoretical bounds are far from being practical. We present an improved implementation of the algorithm, for which unfortunately no worst case bounds are know, but which performs very well in practice. With the modifications presented, the algorithm performs an order of magnitude faster for certain submodular functions.