Monotonic Decompositions of Submodular Set Functions

TL;DR

This paper extends the theory of submodular set functions to an analytic setting, exploring their decompositions, dualities, and subclasses, with applications to graph parameters like cuts and packings.

Contribution

It generalizes finite-set submodular function decompositions to infinite domains and introduces new classes with structural properties, connecting to graph theory.

Findings

Decomposition of submodular functions into nonnegative and increasing parts.

Characterization of infinite-alternating set functions as sums or differences of submodular functions.

Application to weighted cut functions of graphs and related parameters.

Abstract

Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they found applications in the theory of finitely additive measures, nonlinear integrals, and electric capacities. Recently, a number of connections between these two branches have been established, and the aim of this paper is to generalize further results on submodular set functions on finite sets to the analytic setting. We first extend the notion of duality of matroids to submodular set functions, and characterize the uniquely determined decomposition of a submodular set function into the sum of a nonnegaive charge and an increasing submodular set function in which the charge is maximal. Then, we describe basic properties of infinite-alternating set functions, a subclass of submodular set functions that serves as an analytic counterpart of coverage functions. By relaxing the monotonicity assumption in the definition, we introduce a new class of submodular functions with distinguished structural properties that includes, among others, weighted cut functions of graphs. We prove that, unlike general submodular set functions over an infinite domain, any infinite-alternating set function can be written as the sum of an increasing and a decreasing submodular function or as the difference of two increasing submodular functions, thus giving extension of results on monotonic decompositions in the finite case. Finally, motivated by its connections to graph parameters such as the maximum size of a cut and the maximum size of a fractional triangle packing, we study the structure of such decompositions for weighted cut functions of undirected graphs.