Sum of Squares Submodularity
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arXiv
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| Format: | Preprint |
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2025
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| author | Deza, Anna Hall, Georgina |
| author_facet | Deza, Anna Hall, Georgina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24550 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum of Squares Submodularity Deza, Anna Hall, Georgina Optimization and Control Algebraic Geometry Functional Analysis 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. |
| title | Sum of Squares Submodularity |
| topic | Optimization and Control Algebraic Geometry Functional Analysis |
| url | https://arxiv.org/abs/2510.24550 |