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

Symmetric Submodular Function Minimization Under Hereditary Family Constraints