Symmetric Submodular Functions, Uncrossable Functions, and Structural Submodularity

TL;DR

This paper explores the properties of pliable set families satisfying structural submodularity, demonstrating their distinctness from symmetric submodular functions and uncrossable families, with implications for combinatorial optimization.

Contribution

It constructs pliable families that satisfy structural submodularity but are not representable by symmetric submodular functions or decomposable into uncrossable families.

Findings

Constructed pliable families satisfying structural submodularity

Showed these families cannot be represented by symmetric submodular functions

Proved they cannot be partitioned into a small number of uncrossable families

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.