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( 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 function.
Goemans and Soto REF consider the problem of minimizing a symmetric submodular function subject to arbitrary matroid constraints.
Symmetric Submodular Function Minimization Under Hereditary Family Constraints
SIAM J. Discrete Math., 27(2), 1123--1145. 2013
Mathematics Computer Science