Quasi-concave functions on antimatroids

Abstract

In this paper we consider quasi-concave set functions defined on antimatroids. There are many equivalent axiomatizations of antimatroids, that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. An algorthmic characterization of antimatroids, that considers them as set systems, was given in (Kempner, Levit 2003). This characterization is based on the idea of optimization using set functions defined as minimum values of linkages between a set and the elements from the set complement. Such set functions are quasi-concave. Their behavior on antimatroids was studied in (Kempner, Muchnik 2003), where they were applied to constraint clustering. In this work we investigate a duality between quasi-concave set functions and linkage functions. Our main finding is that quasi-concave set functions on an antimatroid may be represented as minimum values of some monotone linkage functions.