Choquet extension of non-monotone submodular setfunctions
László Lovász
TL;DR
The paper addresses extending Choquet's nonlinear integral to all submodular setfunctions on set-algebras, including non-monotone cases, and proves that the corresponding extension $\widehat{\varphi}$ remains convex. It shows that every bounded submodular setfunction has bounded variation, enabling a canonical decomposition $\varphi=\mu-\nu$ and the definition $\widehat{\varphi}=\widehat{\mu}-\widehat{\nu}$, with convexity equivalent to submodularity. A lopsided version of Fubini's theorem is established under smoothness assumptions, yielding an inequality $\widehat{\varphi}(g) \le \int_I \widehat{\varphi}(F_x)\,d\lambda(x)$ for appropriate $F$. The results extend Choquet theory to set-algebras, linking analytic and combinatorial submodularity and informing potential limit theories for matroids and related structures. They provide a rigorous foundation for convex nonlinear functionals arising from submodular setfunctions beyond the monotone setting.
Abstract
In a seminal paper, Choquet introduced an integral formula to extend a monotone increasing setfunction on a sigma-algebra to a (nonlinear) functional on bounded measurable functions. The most important special case is when the setfunction is submodular; then this functional is convex (and vice versa). In the finite case, an analogous extension was introduced by this author; this is a rather special case, but no monotonicity was assumed. In this note we show that Choquet's integral formula can be applied to all submodular setfunctions, and the resulting functional is still convex. We extend the construction to submodular setfunctions defined on a set-algebra (rather than a sigma-algebra). The main property of submodular setfunctions used in the proof is that they have bounded variation. As a generalization of the convexity of the extension, we show that (under smoothness conditions) a ``lopsided'' version of Fubini's Theorem holds.