Just-in-Time Hierarchical Constraint Decomposition

Lazy Clause Generation (LCG) solvers dominate the current constraint programming competitions. These solvers successfully combine systematic propagation based search, global constraints and conflict clause learning from SAT solving into a hybrid approach. My research project extends the LCG methodology by using a mix of eager and lazy encodings and a richer set of constraint decompositions. Global Constraints exhibit a whole hierarchy of different decomposition into more basic constraints. In our work we want to take advantage of such hierarchies and identify criteria on how constraints could be decomposed before and during search. Lazy Clause Generation and Extensions Boolean Satisfiability Solving (SAT) and finite domain Constraint Programming (CP) solve hard combinatorialstructured problems. Pracical SAT solving emerged from the Model Checking and Verification community and constraint programming has some of its success in solving scheduling problems. One attempt to integrate both paradigms into a hybrid solver came with the idea of LCG solvers by (Ohrimenko, Stuckey, and Codish 2009). Early LCG systems collect a clause for each domain filtering of a constraint propagator which can then be used in SAT-style conflict clause resolution. The work of (Abı́o and Stuckey 2012; Abı́o et al. 2013) extends this approach to reactively decompose Pseudo-Boolean constraints lazily during search. If the number of generate clauses during search exceeds a certain limit they decompose to a compact encoding using auxiliary variables. We want to follow up on this strategy and extend it to other constraints. Constraint decomposition faces two challenges: Firstly, the size of the decomposition might be to large. This is especially true in case of decompositions to conjunctive normal form (CNF). Encodings of size O(n) in the domain size of integer variables might already exceed moderate size restrictions. Secondly, the decomposition loses the global view on the constraint and this might hinder propagation, i.e. not all inferences can be found by the filtering algorithms on the decomposed constraints. It is not surprising that larger decomCopyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. positions tend to enforce higher consistency (more propagation). Thus, the key to a good decomposition is a good trade-off between size and consistency. Contrary to the constraint decomposition approach we have the space-efficient propagation algorithms that often achieve higher consistency than a decomposition. Especially when domains are large, eager CNF compilation often exceed reasonable size restrictions. The drawback with pure propagation is the lack of auxiliary variables that often benefit the learning mechanism. To find the right balance between global constraint propagation and decomposition we want to take advantage of a fine-grained view on different levels of decomposition. Constraints can be decomposed into other simpler global constraints that maintain a middle way of representation size and consistency. Furthermore, such extensive analysis will be a fruitful source for auxiliary variables for learning beneficial clauses. The lazy decomposition approach profits from extensive knowledge in constraint decompositions and SAT encodings. The next step in this direction of research is to target a healthy mix of global constraint propagation, explanations for conflict clause learning and decompositions into both non-clausal constraints and CNF. In the next section we will show such a hierarchy by listing decompositions of the ALL-DIFFERENT constraint. In addition to decomposition hierarchies, we will consider various criteria to support the decision of when to decompose and when to propagate such constraints (due to space limitation not mentioned in this extended abstract). We coin the name Just in Time Hierarchical Constraint Decomposition (JIT-HCD) for this methodology. A Concrete Example: ALL-DIFFERENT The constraint ALL-DIFFERENT([x1, . . . xn]) enforces that the values taken by the integer variables are all different, i.e. that xi 6= xj for i < j. This is a well studied constraint in the CP community and many filtering algorithms and decompositions are known. In this section we describe the hierarchy of ALL-DIFFERENT decompositions. The standard domain representation in LCG solvers uses the Boolean variables [[x = v]] and [[x ≥ v]] for each value v ∈ D(x) in the domain of variable x, with [[x ≤ v]] ⇔ ¬[[x ≥ v + 1]]. Let D be union of all domains. The following decompositions are direct translations of the definition of ALL-DIFFERENT: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence