Strongly polynomial separation for the minimum of multiple submodular functions

Develop a strongly polynomial-time algorithm for the separation problem of the sublinear envelope of the pointwise minimum of K submodular functions when K is unrestricted, extending beyond the two-function case addressed in the paper.

Background

The paper reduces separation of the sublinear envelope of a pointwise minimum of K submodular functions to weighted polymatroid intersection over the intersection of K extended polymatroids. For K = 2, the authors obtain a strongly polynomial-time algorithm using weighted polymatroid intersection and lifting techniques.

For general K, the paper provides a compact linear-programming formulation when each individual submodular function has a compact epigraph formulation. However, the authors explicitly note that a strongly polynomial-time algorithm is not known for the general intersection of K polymatroids. Establishing such an algorithm would extend the paper’s tractability results from two disjunctive components to an arbitrary number of components.