Minimizing Submodular Functions over Families of Sets

We consider the problem of characterizing the minimum of a submodular function when the minimization is restricted to a family of subsets. We show that, for many interesting cases, there exist two elements a and b of the groundset such that the problem is equivalent to the problem of minimizing the submodular function over the sets containing a but not b. This leads to a polynomial-time algorithm for minimizing a submodular function over these families of sets. Our results apply, for example, to the families of odd cardinality subsets or even cardinality subsets separating two given vertices, or to the complement of a lattice family of subsets. We also derive that the second smallest value of a submodular function over a lattice family can be computed in polynomial-time. These results generalize and unify several known results.