Interval propagation to reason about sets: Definition and implementation of a practical language

Abstract

Local consistency techniques have been introduced in logic programming in order to extend the application domain of logic programming languages. The existing languages based on these techniques consider arithmetic constraints applied to variables ranging over finite integer domains. This makes difficult a natural and concise modelling as well as an efficient solving of a class of A/P-complete combinatorial search problems dealing with sets. To overcome these problems, we propose a solution which consists in extending the notion of integer domains to that of set domains (sets of sets). We specify a set domain by an interval whose lower and upper bounds are known sets, ordered by set inclusion. We define the formal and practical framework of a new constraint logic programming language over set domains, called Conjunto. Conjunto comprises the usual set operation symbols (U, N, \), and the set inclusion relation (___). Set expressions built using the operation symbols are interpreted as relations (s U Sl = s2 .... ). In addition, Conjunto provides us with a set of constraints called graduated constraints (e.g. the set cardinality) which map sets onto arithmetic terms. This allows us to handle optimization problems by applying a cost function to the quantifiable, i.e., arithmetic, terms which are associated to set terms. The constraint solving in Conjunto is based on local consistency techniques using interval reasoning which are extended to handle set constraints. The main contribution of this paper concerns the formal definition of the language and its design and implementation as a practical language.

Combining Two Structured Domains for Modeling Various Graph Matching Problems

“…and maps, [16], can be used for modeling and solving a wide spectrum of of graph matching problems with any combination of the following properties : monomorphism or isomorphism, graph or subgraph matching, exact or All solutions; subgraph monomorphism over undirected graphs (5 approximate matching (user-specified approximation [9]). To achieve this, we needed to generalize the map variables with non-fixed source and target sets (of the Cardinal kind [31]).…”

“…in [18,19]). CP(Graph) [13] unifies those models by recognizing a common structure: Graph variables are variables whose domain ranges over a set of graphs and as with set variables [20,16], this set of graphs is represented by a graph interval [D(G), D(G)] where D(G), the greatest lower bound (glb) and D(G), the least upper bound (lub) are two graphs with D(G) a subgraph of D(G) (we write D(G) ⊆ D(G)). These two bounds are referred to as the lower and the upper bound.…”

“…The domain of a map variable is thus a set of mappings. Map variables were first introduced in CP in [16] where Gervet defines relation variables. However, the domain and the range of the relations were limited to ground finite sets.…”

“…CP(Graph) can be used to express and solve combinatorial graph problems modeled as constrained subgraph extraction problems. In [16,17], function variables are proposed, but the domain and range are limited to ground sets. Such function variables are useful for modeling problems such as warehouse location.…”

Set Variables and Local Search

“…Most of the popular CP systems of today have features for modelling problems using set variables, i.e., variables taking values that are subsets of some universe. Consider the work by Gervet [6,7], Müller and Müller [13], and Puget [16].…”