Graphic Submodular Function Minimization: A Graphic Approach and Applications

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.