  We present an efficient algorithm to find non-empty minimizers of a symmetric
submodular function over any family of sets closed under inclusion. This for
example includes families defined by a cardinality constraint, a knapsack
constraint, a matroid independence constraint, or any combination of such
constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular
function where $n$ is the cardinality of the ground set. In contrast, the
problem of minimizing a general submodular function under a cardinality
constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina
and Fleischer [2008]).
  The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to
find all nontrivial inclusionwise minimal minimizers of a symmetric submodular
function over a set of cardinality $n$ using $O(n^3)$ oracle calls. Their
procedure in turn is based on Queyranne's algorithm [1998] to minimize a
symmetric submodular
