Polymatroids and Submodularity

Many polytime algorithms have now been presented for minimizing a submodular function f(S) over the subsets S of a finite set E. We provide a tutorial in (somewhat hidden) theoretical foundations of them all. In particular, f can be easily massaged to a set function g(S) which is submodular, non-decreasing, and zero on the empty set, so that minimizing f(S) is equivalent to repeatedly determining whether a point x is in the polymatroid, P(g) = {x : x >= 0 and, for every S, sum of x(j) over j in S is at most g(S)}. A fundamental theorem says that, assuming g(S) is integer, the 0,1 vectors x in P(g) are the incidence vectors of the independent sets of a matroid M(P(g)). Another gives an easy description of the vertices of P(g). We will show how these ideas provide beautiful, but complicated, polytime algorithms for the possibly useful optimum branching system problem.