Modular lazy search for Constraint Satisfaction Problems

Abstract

We describe a unified, lazy, declarative framework for solving constraint satisfaction problems, an important subclass of combinatorial search problems. These problems are both practically significant and computationally hard. Finding solutions involves combining good general-purpose search algorithms with problem-specific heuristics. Conventional imperative algorithms are usually implemented and presented monolithically, which makes them hard to understand and reuse, even though new algorithms often are combinations of simpler ones. Lazy functional languages, such as Haskell, encourage modular structuring of search algorithms by separating the generation and testing of potential solutions into distinct functions communicating through an explicit, lazy intermediate data structure. But only relatively simple search algorithms have been treated this way in the past. Our framework uses a generic generation and pruning algorithm parameterized by a labeling function that annotates search trees with conflict sets. We show that many advanced imperative search algorithms, including conflict-directed backjumping, backmarking, minimal forward checking, and fail-first dynamic variable ordering, can be obtained by suitable instantiation of the labeling function. More importantly, arbitrary combinations of these algorithms can be built by simply composing their labeling functions. Our modular algorithms are as efficient as the monolithic imperative algorithms in the sense that they make the same number of consistency checks, and most of our algorithms are within a constant factor of their imperative counterparts in runtime and space usage. We believe our framework is especially well-suited for experimenting to find good combinations of algorithms for specific problems.