On the Supremum of Singleton Ratios in Submodular Functions

TL;DR

This paper investigates the maximum influence a single element can have on submodular functions, establishing bounds that grow with the set size and highlighting open problems in narrowing this gap.

Contribution

The authors construct an example showing the influence measure can be as large as Omega(n/log n) and prove a doubly exponential upper bound, advancing understanding of variable influence in submodular functions.

Findings

Constructed an example with influence as large as Omega(n/log n).

Established a doubly exponential upper bound on the influence measure.

Identified the open problem of narrowing the gap between bounds.

Abstract

Let $N$ be a finite set of cardinality $n$, and $a\in N$. A submodular function $f$ on $N$ with $f(a)=1$ is defined to be $a$-reduced if, for any decomposition $f=g+h$ into submodular functions where $h$ does not depend on $a$, it follows that $h$ is identically zero. The maximal possible value of $f$ on the remaining singletons defines a quantity $\lambda$ that characterizes the degree to which one variable can constrain the value of another; geometrically, it also limits the possible elongation of the associated submodular base polytope. We construct an example demonstrating that $\lambda$ can be as large as $\Omega(n/\log n)$. Furthermore, we establish a doubly exponential upper bound on $\lambda$. The problem of narrowing the gap between these bounds remains open.