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 $λ$ 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 $λ$ can be as large as $Ω(n/\log n)$. Furthermore, we establish a doubly exponential upper bound on $λ$. The problem of narrowing the gap between these bounds remains open.

翻译：设 $N$ 为基数为 $n$ 的有限集，且 $a\in N$。定义在 $N$ 上满足 $f(a)=1$ 的子模函数 $f$ 为 $a$-约化的，若对于任意分解 $f=g+h$（其中 $g$ 和 $h$ 为子模函数，且 $h$ 不依赖于 $a$），必有 $h$ 恒为零。其余单例上 $f$ 的最大可能取值定义了一个量 $\lambda$，该量刻画了一个变量对另一个变量值的约束程度；从几何角度而言，它也限制了关联子模基多面体的可能伸长。我们构造了一个例子，证明 $\lambda$ 可达到 $\Omega(n/\log n)$ 的量级。此外，我们建立了 $\lambda$ 的双重指数上界。缩小这些界之间差距的问题仍未解决。