Tree Search and ARC Consistency in Constraint Satisfaction Algorithms

Abstract

Constraint satisfaction problems are ubiquitous in Artificial Intelligence and many algorithms have been developed for their solution. This paper provides a unified introduction to some of these algorithms, including Backtracking, Haralick’s Forward Checking, Partial Lookahead and Full Lookahead, and three based on the arc-consistency algorithms that Mackworth has called AC1, AC2 and AC3. It is shown that these can all be unified as having the common structure of tree search (TS) augmented with arc-consistency algorithms (AC) of various extent. Haralick’s algorithms, and even traditional Backtracking, are seen to contain a partial-arc-consistency component. In analogy to Mackworth’s nomenclature these are named AC1/5, AC1/4, AC1/3 and AC1/2 — the fractional suffix being more or less proportional to the degree of arc-consistency attained. The algorithms may then be unified as being of the form TS + ACi 1 or TS + ACi 1 + ACi 2, for various fractional and integer i 1 and i 2. A combined algorithm based on this structure is presented and algorithm efficiencies are compared empirically, using the n-queens problem and a new version called confused n-queens. We find that it may very well pay to trade more tree search for a reduction in arc consistency — that is, to allow a larger search tree (in terms of nodes), as a result of less arc consistency at the nodes, in order that the overall effort (in terms of constraint checks) be reduced.