Sum of Squares Submodularity

TL;DR

This paper introduces a hierarchy of algebraic certificates called t-sos submodularity for certifying submodularity of set functions, enabling polynomial-time verification and applications in optimization.

Contribution

It develops a new hierarchy of algebraic conditions for submodularity, with polynomial-time verification, and demonstrates applications in optimization and algebraic characterization.

Findings

Polynomial-size semidefinite programs verify t-sos submodularity.

Identifies cases where submodularity and t-sos submodularity coincide.

Provides applications in submodular regression and optimization bounds.

Abstract

We introduce the notion of $t$-sum of squares (sos) submodularity, which is a hierarchy, indexed by $t$, of sufficient algebraic conditions for certifying submodularity of set functions. We show that, for fixed $t$, each level of the hierarchy can be verified via a semidefinite program of size polynomial in $n$, the size of the ground set of the set function. This is particularly relevant given existing hardness results around testing whether a set function is submodular (Crama, 1989). We derive several equivalent algebraic characterizations of $t$-sos submodularity and identify submodularity-preserving operations that also preserve $t$-sos submodularity. We further present a complete classification of the cases for which submodularity and $t$-sos submodularity coincide, as well as examples of $t$-sos-submodular functions. We demonstrate the usefulness of $t$-sos submodularity through three applications: (i) a new convex approach to submodular regression, involving minimal manual tuning; (ii) a systematic procedure to derive lower bounds on the submodularity ratio in approximate submodular maximization, and (iii) improved difference-of-submodular decompositions for difference-of-submodular optimization. Overall, our work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.