Article

# A Fast Algorithm for Finding Matching Responses in a Survey Data Table

Byrej 269, 2650 Hvidovre, Copenhagen, Denmark

Mathematical Social Sciences (Impact Factor: 0.45). 03/1995; 30:195-205. DOI: 10.1016/0165-4896(95)00780-P Source: CiteSeer

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**ABSTRACT:**This paper deals with maximization of set functions defined 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. In this paper, we extend this finding to meet-semilattices.We show that the class of functions defined as minimum values of monotone linkage functions coincides with the class of quasi-concave set functions. Quasi-concave functions determine a chain of upper level sets each of which is a meet-semilattice. This structure allows development of a polynomial algorithm that finds a minimal set on which the value of a quasi-concave function is maximum. One of the critical steps of this algorithm is a set closure. Some examples of closure computation, in particular, a closure operator for convex geometries, are considered.Discrete Applied Mathematics. 01/2008; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this article, we have investigated monotone linkage functions defined on convex geometries, antimatroids, and semilattices in general. It has been shown that the class of functions defined as minimum values of monotone linkage functions has close relationship with the class of quasi-concave set functions. Quasi-concave functions defined on semilattices, antimatroids and convex geometries determine special substructures of these set families. This structures allow building efficient algorithms that find minimal sets on which values of quasi-concave functions are maximum. The mutual critical step of these algorithms is how to describe the set closure operator. If an efficient algorithm of the closure construction exists it causes the optimization algorithm to be efficient as well. On the other hand, we think that the closure construction problem is interesting enough to be investigated separately. Thus, we suppose that for an arbitrary semilattice the problem of closure construction has exponential complexity. An interesting direction for future work is to develop our methods for relational databases, where a polynomial algorithm for closure construction is known [3]. We have considered some applications of quasi-concave functions to clustering a structured data set, where together with pair-wise similarities between objects, we are also given additional information about objects organization. We focused on a simple structure - a partition model of data where the objects are a priori partitioned into groups. While clustering such data, we also considered an additional requirement of being able to differentiate between pairwise similarities across different partite sets. Existing clustering methods do not solve this problem, since they are limited to finding clusters in a collection of isolated objects. The requirement of differentially treating pair-wise relationships across different groups was modeled by a multipartite graph along with a hierarchical relationship between these groups. The problem was reduced to finding the cluster (subgraph) of the highest density in the multipartite graph.11/2008; , ISBN: 978-953-7619-27-5 - [Show abstract] [Hide abstract]

**ABSTRACT:**Now it is almost a common truth that measures of the government to avoid damage or contamination of the environment become significant for nature protection efforts and wild life preservation. Sometimes, possible outcomes of such efforts might turn into voluntary solutions that certain companies are keyed up to find a break in the legislation not to, or only formally, participate in environment protection. In the situation, which takes into account the nature/content of human behavior, we blow the perception of the reality out of all proportions. However, the reader may find the interaction of interests of companies and environment protection agency informative. One may also think that an efficiency of a solution is powerful enough of being right and proper. Nevertheless, advancing in other direction of democratic, i.e., selfgoverning decision-making process, we still have a chance to make a point.10/2009;

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