Symmetric Submodular Functions, Uncrossable Functions, and Structural Submodularity

2025-12-31

arXiv.org

AI Summary

This study investigates the interplay among structural submodularity, symmetric submodular functions, and cross-free families, addressing whether every structurally submodular flexible set family can either be represented as a lower level set of some symmetric submodular function or partitioned into finitely many cross-free families. By synthesizing tools from combinatorial set theory and the theory of abstract separation systems, the authors construct a novel class of structurally submodular flexible set families that provably cannot be expressed as lower level sets of any symmetric submodular function nor decomposed into any finite number—indeed, not even into any prescribed number \(d \geq 2\)—of cross-free families. This counterexample demonstrates a fundamental distinction among these three concepts, refuting conjectured equivalences and underscoring the intrinsic complexity and independence of structural submodularity.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Nonmonotonic ReasoningMachine Learning: Structured Learning

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Bridging structured and unstructured dataSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail, and Vazirani (Combinatorica 15 (1995)) call a family of sets $\mathcal{F}$ uncrossable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, both $A\cap{B},A\cup{B}$ are in $\mathcal{F}$, or both $A-B,B-A$ are in $\mathcal{F}$. Bansal, Cheriyan, Grout, and Ibrahimpur (Algorithmica 86 (2024), arXiv:2209.11209) call a family of sets $\mathcal{F}$ pliable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, at least two of the sets $A\cap{B},A\cup{B},A-B,B-A$ are in $\mathcal{F}$. We say that a pliable family of sets $\mathcal{F}$ satisfies structural submodularity if the following holds: for any pair of crossing sets $A,B\in\mathcal{F}$, at least one of the sets $A\cap{B},A\cup{B}$ is in $\mathcal{F}$, and at least one of the sets $A-B,B-A$ is in $\mathcal{F}$. For any positive integer $d\geq2$, we construct a pliable family of sets $\mathcal{F}$ that satisfies structural submodularity such that (a) there do not exist a symmetric submodular function $g$ and $\lambda\in{\mathbb Q}$ such that $\mathcal{F} = \{ S \,:\, g(S)<\lambda \}$, and (b) $\mathcal{F}$ cannot be partitioned into $d$ (or fewer) uncrossable families.

Problem

• symmetric submodular functions
• uncrossable families
• structural submodularity
• pliable families
• set systems

Innovation

• structural submodularity
• pliable family
• symmetric submodular function
• uncrossable family
• abstract separation systems