Extending Compact-Table to Negative and Short Tables

Table constraints are very useful for modeling combinatorial constrained problems, and thus play an important role in Constraint Programming (CP). During the last decade, many algorithms have been proposed for enforcing the property known as Generalized Arc Consistency (GAC) on such constraints. A state-of-the art GAC algorithm called CompactTable (CT), which has been recently proposed, significantly outperforms all previously proposed algorithms. In this paper, we extend this algorithm in order to deal with both short supports and negative tables, i.e., tables that contain universal values and conflicts. Our experimental results show the interest of using this fast general algorithm. Introduction Table constraints, also called extension(al) constraints, explicitly express for the variables they involve, either the allowed combinations of values, called supports, or the forbidden combinations of values, called conflicts. Table constraints can theoretically encode any kind of restrictions and are consequently very important in Constraint Programming (CP). Indeed, as especially claimed by people from industry (e.g., IBM and Google), table constraints are often required when modeling combinatorial constrained problems in many application fields. The design of filtering algorithms for such constraints has generated a lot of research effort, see (Bessiere and Régin 1997; Lhomme and Régin 2005; Lecoutre and Szymanek 2006; Gent et al. 2007; Ullmann 2007; Lecoutre 2011; Lecoutre, Likitvivatanavong, and Yap 2015; J.-B. Mairy and Deville 2014; Perez and Régin 2014; Wang et al. 2016; Demeulenaere et al. 2016). On classical tables, i.e., sequences of ordinary tuples, the algorithmic progresses that have been made over the years for maintaining the property called GAC (Generalized Arc Consistency) are quite impressive. Roughly speaking, an algorithm such as Compact-Table (Demeulenaere et al. 2016) is about one order of magnitude faster than the best algorithm(s) proposed a decade ago (Lhomme and Régin 2005; Lecoutre and Szymanek 2006; Gent et al. 2007; Ullmann 2007). Unfortunately, table constraints admit practical boundaries because the memory space required to represent them may grow exponentially with their arity. To reduce Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. space complexity, researchers have focused on various forms of compression. For example, tries (Gent et al. 2007), Multivalued Decision Diagrams (MDDs) (Cheng and Yap 2010; Perez and Régin 2014) and Deterministic Finite Automaton (DFA) (Pesant 2004) are general structures used to represent table constraints in a compact way, so as to facilitate filtering process. Cartesian product is another classical mechanism to represent compactly large sets of tuples. This is the approach followed by works on compressed tuples (Katsirelos and Walsh 2007; Régin 2011; Xia and Yap 2013) and short supports and tuples (Jefferson and Nightingale 2013). A short tuple allows the presence of universal values, denoted by the symbol *, meaning that some variables can take any values from their domains. Other forms of compact representation are obtained by means of sliced tables (Gharbi et al. 2014) and smart tables (Mairy, Deville, and Lecoutre 2015). Compact-Table (CT) is a state-of-the-art GAC algorithm for positive (ordinary) table constraints, i.e., constraints defined by tables containing (uncompressed) supports. In this paper, we extend CT in order to be able to deal with: • negative tables (i.e., tables containing conflicts), • and/or short tuples (i.e., tuples containing the symbol *). Technical Background A constraint network (CN)N is composed of a set of n variables and a set of e constraints. Each variable x has an associated domain, denoted by dom(x), that contains the finite set of values that can be assigned to it. Each constraint c involves an ordered set of variables, called the scope of c and denoted by scp(c), and is semantically defined by a relation, denoted by rel(c), which contains the set of tuples allowed for the variables involved in c. The arity of a constraint c is |scp(c)|. For simplicity, a variable-value pair (x, a) such that x ∈ scp(c) and a ∈ dom(x) is called a value (of c). A table constraint c is a constraint such that rel(c) is defined explicitly by listing (in a table) the tuples that are allowed by c or the tuples that are disallowed by c. In the former case, the table constraint is said to be positive whereas in the latter case, it is said negative. Let τ = (a1, a2, . . . , ar) be a tuple of values associated with an ordered set of variables vars(τ) = {x1, x2, . . . , xr}. The ith value of τ is denoted by τ [i] or