Sum of Squares Submodularity

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 results by Crama around testing whether a set function is submodular. 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.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2025.2422.