Algorithms for Symmetric Submodular Function Minimization under Hereditary Constraints and Generalizations

We present an efficient algorithm to find nonempty minimizers of a symmetric submodular function $f$ over any family of sets ${\cal I}$ closed under inclusion. Our algorithm makes $O(n^3)$ oracle calls to $f$ and ${\cal I}$, 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 $o(\sqrt{n/\log n})$ [Z. Svitkina and L. Fleischer, in Proceedings of the $49$th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Washington, DC, 2008, pp. 697--706]. We also present two extensions of the above algorithm. The first extension reports all nontrivial inclusionwise minimal minimizers of $f$ over ${\cal I}$ using $O(n^3)$ oracle calls, and the second reports all extreme subsets of $f$ using $O(n^4)$ oracle calls. Our algorithms are similar to a procedure by Nagamochi and Ibaraki [Inform. Process. Lett., 67 (1998), pp. 239--244] that finds all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of size $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [M. Queyranne, Math. Program., 82 (1998), pp. 3--12] to minimize a symmetric submodular function by finding pendent pairs. Our results extend to any class of functions for which we can find a pendent pair whose head is not a given element. [PUBLICATION ABSTRACT]