Graphic Submodular Function Minimization: A Graphic Approach and Applications

Summary

In this paper we study particular submodular functions that we call “graphic”. A graphic submodular function is defined on the edge set E of a graph G=(V,E) and is equal to the sum of the rank-function of G and of a linear function on E. Several polynomial algorithms are known that can be used to minimize graphic submodular functions and some were adapted to an equivalent problem called “Optimal Attack” by Cunningham. We collect eight different algorithms for this problem, including a recent one (initially developed for solving a problem for physics): it consists of |V|−1 steps, where the i-th step requires the solution of a network flow problem on a subgraph (with slight modifications) induced by at most i vertices of the given graph (i=2,…,|V|). This is a fully combinatorial algorithm for this problem: contrary to its predecessors, neither the algorithm nor its proof of validity use directly linear programming or keep any kind of dual solution. The approach is direct and conceptually simple, with the same worst case asymptotic complexity as the previous ones. Motivated by applications, we also show how this combinatorial approach to graphic submodular function minimization provides efficient solution methods for several problems of combinatorial optimization and physics.

In honor of Bernhard Korte’s 70-th birthday and in memory of the pioneering work and important achievements that have been reached concerning submodular functions in the Institut für Disckete Mathematik, Ökonometrie und Operations Research in the ’80-s and ’90-s, due to the generous visitor’s program and high quality research and training of the institute led by Professor Korte, the results of which provide the background of the present work.