Symmetric Submodular Function Minimization Under Hereditary Family Constraints

TL;DR

This paper introduces an efficient $O(n^3)$ oracle call algorithm for minimizing symmetric submodular functions over hereditary families, including various constrained set families, outperforming known inapproximability bounds.

Contribution

The paper presents a novel algorithm for symmetric submodular minimization under hereditary constraints, extending previous methods to a broader class of set families with improved efficiency.

Findings

Algorithm makes $O(n^3)$ oracle calls.

Applicable to families defined by cardinality, knapsack, and matroid constraints.

Outperforms inapproximability bounds for general submodular minimization.

Abstract

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