On the Supremum of Singleton Ratios in Submodular Functions

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.