Minimizing Submodular Functions over Hierarchical Families

TL;DR

This paper demonstrates polynomial-time algorithms for submodular function minimization over certain hierarchical family complements, advancing understanding of tractability in structured subset families.

Contribution

It introduces polynomial-time solutions for SFM over complements of hierarchical families, including intersecting, crossing, and lattice unions, partially resolving an open problem.

Findings

Polynomial-time SFM over complements of hierarchical families.

Polynomial-time computation of the k-th smallest submodular value for constant k.

Partial resolution of the open problem on SFM over intersection of parity families.

Abstract

This paper considers submodular function minimization (SFM) restricted to a family of subsets. We show that SFM over complements of families with certain hierarchical structures can be solved in polynomial-time. This yields a polynomial-time algorithm for SFM over complements of various families, such as intersecting families, crossing families, and the unions of lattices. Moreover, this tractability result partially settles the open question posed by N\"agele, Sudakov, and Zenklusen on polynomial-solvability of SFM over the intersection of parity families. Furthermore, our tractability result implies that for a constant positive integer $k$, the $k$-th smallest value of a submodular function can be obtained in polynomial-time.