The CLP Procedure

A more efficient technique than backtracking is that of constraint propagation, which uses consistency techniques to effectively prune the domains of variables. Consistency techniques are based on the idea of a priori pruning, which uses the constraint to reduce the domains of the variables. Consistency techniques are also known as relaxation algorithms (Tsang, 1993), and the process is also referred to as problem reduction, domain filtering, or pruning.

One of the earliest applications of consistency techniques was in the AI field in solving the scene labeling problem, which required recognizing objects in three-dimensional space by interpreting two-dimensional line drawings of the object. The Waltz filtering algorithm (Waltz, 1975) analyzes line drawings by systematically labeling the edges and junctions while maintaining consistency between the labels.

An effective consistency technique for handling resource capacity constraints is edge finding (Applegate and Cook, 1991). Edge-finding techniques reason about the processing order of a set of activities that require a given resource or set of resources. Some of the earliest work related to edge finding can be attributed to Carlier and Pinson (1989), who successfully solved MT10, a well-known 10×10 job shop problem that had remain unsolved for over 20 years (Muth and Thompson, 1963).

Constraint propagation is characterized by the extent of propagation (also referred to as the level of consistency) and the domain pruning scheme that is followed: domain propagation or interval propagation. In practice, interval propagation is preferred over domain propagation because of its lower computational costs. This mechanism is discussed in detail in Van Hentenryck (1989). However, constraint propagation is not a complete solution technique and needs to be complemented by a search technique in order to ensure success (Kumar, 1992).

Finite-domain constraint programming is an effective and complete solution technique that embeds incomplete constraint propagation techniques into a nondeterministic backtracking search mechanism, implemented as follows. Whenever a node is visited, constraint propagation is carried out to attain a desired level of consistency. If the domain of each variable reduces to a singleton set, the node represents a solution to the CSP. If the domain of a variable becomes empty, the node is pruned. Otherwise a variable is selected, its domain is distributed, and a new set of CSPs is generated, each of which is a child node of the current node. Several factors play a role in determining the outcome of this mechanism, such as the extent of propagation (or level of consistency enforced), the variable selection strategy, and the variable assignment or domain distribution strategy.

For example, the lack of any propagation reduces this technique to a simple generate-and-test, whereas performing consistency on variables already selected reduces this to chronological backtracking, one of the systematic search techniques. These are also known as look-back schemas, because they share the disadvantage of late conflict detection. Look-ahead schemas, on the other hand, work to prevent future conflicts. Some popular examples of look-ahead strategies, in increasing degree of consistency level, are forward checking (FC), partial look ahead (PLA), and full look ahead (LA) (Kumar, 1992). Forward checking enforces consistency between the current variable and future variables; PLA and LA extend this even further to pairs of not yet instantiated variables.

Two important consequences of this technique are that inconsistencies are discovered early and that the current set of alternatives that are coherent with the existing partial solution is dynamically maintained. These consequences are powerful enough to prune large parts of the search tree, thereby reducing the “combinatorial explosion” of the search process. However, although constraint propagation at each node results in fewer nodes in the search tree, the processing at each node is more expensive. The ideal scenario is to strike a balance between the extent of propagation and the subsequent computation cost.

Variable selection is another strategy that can affect the solution process. The order in which variables are chosen for instantiation can have a substantial impact on the complexity of the backtrack search. Several heuristics have been developed and analyzed for selecting variable ordering. One of the more common ones is a dynamic heuristic based on the fail first principle (Haralick and Elliot, 1980), which selects the variable whose domain has minimal size. Subsequent analysis of this heuristic by several researchers has validated this technique as providing substantial improvement for a significant class of problems. Another popular technique is to instantiate the most constrained variable first. Both these strategies are based on the principle of selecting the variable most likely to fail and to detect such failures as early as possible.

The domain distribution strategy for a selected variable is yet another area that can influence the performance of a backtracking search. However, good value-ordering heuristics are expected to be very problem-specific (Kumar, 1992).