🤖 AI Summary
This work investigates the maximum possible ratio λ of function values over singleton sets for submodular functions, under the constraint that the function value on a given singleton is fixed to 1. This ratio characterizes the strength of inter-variable constraints and the extremal scaling behavior of base polyhedra. By constructing a novel class of a-reduced submodular functions and integrating techniques from submodular decomposition, extremal combinatorics, and polyhedral geometry, the authors present the first explicit construction achieving a lower bound of λ = Ω(n / log n). They also establish a double-exponential upper bound on λ, thereby revealing a substantial gap between existing theoretical bounds. This result provides a foundational step toward tightening the known bounds on λ in future research.
📝 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 $λ$ 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.