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 I closed under inclusion. Our algorithm makes O(n3) oracle calls to f and 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( √ n/ logn) [Z. Svitkina and L. Fleischer, in Proceedings of the 49th 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 I using O(n3) oracle calls, and the second reports all extreme subsets of f using O(n4) 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(n3) 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.