Abstract
This paper considers additive decompositions by proper monotone set functions (i.e., monotone set functions with proper support), and investigates whether it is possible to have additive decompositions for 2-monotone set functions (i.e., monotone, supermodular set functions), by monotone set functions with proper support which is called a "decomposability question." The authors propose a computational approach by formulating the convex polyhedral cone of 2-monotone set functions, and by examining the generators of the cone for further decomposability over proper monotone set functions. This enables to numerically resolve the decomposability question in a finite steps of procedure. The authors have tested the proposed method for five- and six-element sets, and obtained a positive answer to the question for the five-element set and a negative one for the six-element set.
