A note on the Minimum Norm Point algorithm

We present a provably more efficient implementation of the Minimum Norm Point Algorithm conceived by Fujishige than the one presented in [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 [FUJI82]. It is known that the problem can also be efficiently solved in strongly polynomial time [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. Introduction Given a base set S, a submodular function F is such that, for any A,B ⊆ S the following holds F (A) + F (B) ≥ F (A ∩B) + F (A ∪ B) (1) It is not hard to show, that a cut in the graph is a submodular function, where F (A) = cut(A, V \A). The objective is to minimize the cut and which in turn enables us to find a maximum flow in a graph. It is also known that any symmetric submodular function, that is for F (A) = F (S\A) for all A ⊆ S, can be seen as a cut function in a certain graph [QUER95]. Base Polyhedra and Submodular Function Minimization Throughout this paper we assume that for a set E ∈ 2 and a point x ∈ R x(E) = ∑ ei∈E xei or a sum of projection on coordinates in E. It will also be useful to define a base polyhedron B(F ) with respect to a submodular function F :