Quasi-concave functions on antimatroids
نویسندگان
چکیده
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 [4]. 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 [5], 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 an arbitrary quasi-concave set function on antimatroid may be represented as minimum values of some monotone linkage function. keywords: antimatroid, quasi-concave set function, monotone linkage function.
منابع مشابه
Clustering on antimatroids and convex geometries
The clustering problem as a problem of set function optimization with constraints is considered. The behavior of quasi-concave functions on antimatroids and on convex geometries is investigated. The duality of these two set function optimizations is proved. The greedy type Chain algorithm, which allows to find an optimal cluster, both as the most distant group on antimatroids and as a dense c...
متن کاملDuality between quasi-concave functions and monotone linkage functions
A function F defined on all subsets of a finite ground set E is quasiconcave if F (X∪Y ) ≥ min{F (X), F (Y )} for all X, Y ⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concav...
متن کاملQuasi-Concave Functions and Greedy Algorithms
Many combinatorial optimization problems can be formulated as: for a given set system over E (i.e., for a pair (E, ) where ⊆ 2E is a family of feasible subsets of finite set E), and for a given function F : →R, find an element of for which the value of the function F is minimum or maximum. In general, this optimization problem is NP-hard, but for some specific functions and set systems the prob...
متن کاملErc Sets and Antimatroids
Grammars in Optimality Theory can be characterized by sets of Ercs (Elementary Ranking Conditions). Antimatroids are structures that arose initially in the study of lattices. In this paper we prove that antimatroids and consistent Erc sets have the same formal structures. We do so by defining two functions MChain and RCerc, MChain being a function from consistent sets of Ercs to antimatroids an...
متن کاملQuasi-concave functions on meet-semilattices
This paper deals with maximization of set f'unctions delined as minimum values of monotone linkage functions. In previous research, it has been shown that such a set function can be maximized by a greedy type algorithm over a family of all subsets of a finite set. ln this paper, we extend this finding to meet-semilattices. We show that the class of functions defined as minimum values of monoton...
متن کامل