For some computational problems (e.g., product configuration, planning, diagnosis, query answering, phylogeny reconstruction) computing a set of similar diverse solutions may be desirable for better decision-making. With this motivation, we studied several decision optimization versions of this problem in the context of Answer Set Programming (ASP), analyzed their computational complexity, and introduced offline online methods to compute similar diverse solutions of such computational problems with respect to a given distance function. All these methods rely on the idea of computing solutions to a problem by means of finding the answer sets for an ASP program that describes the problem. The offline methods compute all solutions in advance using the ASP formulation of the problem with an ASP solver, like Clasp, and then identify similar diverse solutions using clustering methods. The online methods compute similar diverse solutions following one of the three approaches: by reformulating the ASP representation of the problem to compute similar diverse solutions at once using an ASP solver; by computing similar diverse solutions iteratively (one after other) using an ASP solver; by modifying the search algorithm of an ASP solver to compute similar diverse solutions incrementally. We modified Clasp to implement the last online method and called it Clasp-NK. In the first two online methods, the given distance function is represented in ASP; in the last one it is implemented in C++. We showed the applicability and the effectiveness of these methods on reconstruction of similar diverse phylogenies for Indo-European languages, and on several planning problems in Blocks World. We observed that in terms of computational efficiency the last online method outperforms the others; also it allows us to compute similar diverse solutions when the distance function cannot be represented in ASP.
@cite_31: Users can often naturally express their preferences in terms of ideal or non-ideal solutions. We show how to reason about logical combinations of distance constraints on ideals and non-ideals using a novel global constraint. We evaluate our approach on both randomly generated and real-world configuration problem instances.
@cite_23: It is useful in a wide range of situations to find solutions which are diverse (or similar) to each other. We therefore define a number of different classes of diversity and similarity problems. For example, what is the most diverse set of solutions of a constraint satisfaction problem with a given cardinality? We first determine the computational complexity of these problems. We then propose a number of practical solution methods, some of which use global constraints for enforcing diversity (or similarity) between solutions. Empirical evaluation on a number of problems show promising results.
As for the computational complexity analysis of these problems, the authors find out that they are all -complete; these results comply with the ones presented in subject to conditions under which the problems of @cite_23 @cite_31 above are equivalent to the problems we study in this paper.