On Fundamental Operations for Multimodular Functions

Multimodular functions, primarily used in the literature of queueing theory, discrete-event systems, and operations research, constitute a fundamental function class in discrete convex analysis. The objective of this paper is to clarify the properties of multimodular functions with respect to fundamental operations such as permutation and scaling of variables, projection (partial minimization) and convolution. It is shown, in particular, that the class of multimodular functions is stable under projection under a certain natural condition on the variables to be minimized, and the convolution of two multimodular functions is not necessarily multimodular, even in the special case of the convolution of a multimodular function with a separable convex function.