A constraint satisfaction problem revolves finding values for a set of variables subject to a set of constraints (relations) on those variables Backtrack search is often used to solve such problems. A relationship involving the structure of the constraints is described which characterizes to some degree the extreme case of mimmum backtracking (none) The relationship involves a concept called "width," which may provide some guidance in the representation of constraint satisfaction problems and the order m which they are searched The width concept is studied and applied, in particular, to constraints which form tree structures.

A constraint network representation is presented for a combinatorial search problem: finding values for a set of variables subject to a set of constraints. A theory of consistency levels in such networks is formulated, which is related to problems of backtrack tree search efficiency. An algorithm is developed that can achieve any level of consistency desired, in order to preprocess the problem for subsequent backtrack search, or to function as an alternative to backtrack search by explicitly determining all solutions.