Open Constraints in a Closed World

Abstract

Abstract. We study domain filtering algorithms for open constraints, i.e., constraints that are not a priori defined on specific sets of variables. We present an efficient filtering algorithm, achieving set-domain consistency, for open global cardinality constraints. We extend this result to conjunctions of them, in case they are defined on disjoint sets of variables. We also analyze the case when the sets of variables may overlap. As establishing set-domain consistency is NP-complete in that case, we propose a weaker, though efficient, filtering algorithm instead. Finally, we extend our results to conjunctions of similar open constraints.

Open Packing for Facade-Layout Synthesis Under a General Purpose Solver

“…Basically, our solution follows the approach in [22] where a set variable is used to handle decision variables that are potentially in the solution. In this work however, as rectangle variables are already composed of several integer attributes, we found more natural to use an extra binary variable per rectangle instead of a set variable.…”

“…Note that, as a rectangle is already defined by an array of integer variables (its coordinates and size), it is more natural to extend it with a fifth binary variable representing its presence in the solution than introducing a set variable to represent all present rectangles [22]. Domains for each variable depends on the variable semantics.…”

“…This corresponds to the Open [22] variant of the DiffN [3] constraint, i.e. a generalization of DiffN to handle optional rectangles.…”

“…Nonetheless, even though the idea seems simple, a good implementation is intricate. Instead, our work is inspired from [22], where the authors introduce open global constraints. An open global constraint is an extension of an existing global constraint that includes a set variable (or an array of binary variables) to indicate the subset of decision variables the former constraint holds on.…”

Range and Roots: Two common patterns for specifying and propagating counting and occurrence constraints

“…The Gcc constraint has been extended to OpenGcc, a Gcc constraint for which the set of variables involved is not known in advance [34]. Given variables X 1 , .., X n and a set variable S, ∅ ⊆ S ⊆ {1..n}, OpenGcc([X 1 , .., X n ], S, [d 1 , .., d m ], [O 1 , .., O m ]) holds iff |{i ∈ S | X i = d j }| = O j for all j. OpenGcc can be decomposed into a set of Roots constraints in almost the same way as Gcc was decomposed in Section 6.2.1:…”