In CP literature combinatorial design problems such as sport scheduling, Steiner systems, error-correcting codes and more, are typically solved using Finite Domain (FD) models despite often being more naturally expressed as Finite Set (FS) models. Existing FS solvers have difficulty with such problems as they do not make strong use of the ubiquitous set cardinality information. We investigate a new approach to strengthen the propagation of FS constraints in a tractable way: extending the domain representation to more closely approximate the true domain of a set variable. We show how this approach allows us to reach a stronger level of consistency, compared to standard FS solvers, for arbitrary constraints as well as providing a mechanism for implementing certain symmetry breaking constraints. By experiments on Steiner Systems and error correcting codes, we demonstrate that our approach is not only an improvement over standard FS solvers but also an improvement on recently published results using FD 0/1 matrix models as well. 1 O(ncv √ nc) where n=num vars, c=cardinality and v=size of largest lub 2 See Sect. 6 for explanation of notation

Abstract. Since their beginning in constraint programming, set solvers have been applied to a wide range of combinatorial search problems, such as bin-packing, set partitioning, circuit design, and Combinatorial Design Problems. In this paper we present and evaluate a new means towards improving the practical reasoning power of Finite Set (FS) constraint solvers to better address such set problems with a particular attention to the challenging symmetrical set problems often cast as Combinatorial Design Problems (CDPs). While CDPs find a natural formulation in the language of sets, in constraint programming literature, alternative models are often used due to a lack of efficiency of traditional FS solvers. We first identify the main structural components of CDPs that render their modeling suitable to set languages but their solving a technical challenge. Our new prototype solver extends the traditional subset variable domain with lexicographic bounds that better approximate a set domain by satisfying the cardinality restrictions applied to the variable, and allow for active symmetry breaking using ordering constraints. Our contribution includes the formal and practical framework of the new solver implemented on top of a traditional set solver, and an empirical evaluation on benchmark CDPs.