An Elimination Algorithm for Functional Constraints

Functional constraints are studied in Constraint Satisfaction Problems (CSP) using consistency concepts (e.g., [4, 1]). In this paper, we propose a new method — variable substitution — to process functional constraints. The idea is that if a constraint is functional on a variable, this variable in another constraint can be substituted using the functional constraint without losing any solution. We design an efficient algorithm to reduce, in O(ed), a general binary CSP containing functional constraints into a canonical form which simplifies the problem and makes the functional portion trivially solvable. When the functional constraints are also bi-functional, then the algorithm is linear in the size of the CSP. We use the standard notations in CSP. Two CSPs are equivalent if and only if they have the same solution space. Throughout this paper, n represents the number of variables, d the size of the largest domain of the variables, and e the number of constraints in a problem. The composition of two constraints is defined as cjk ◦ cij = {(a, c) | ∃b ∈ Dj , such that (a, b) ∈ cij ∧ (b, c) ∈ cjk}. Composing cij and cjk gives a new constraint on i and k. A constraint cij is functional on j if for any a ∈ Di there exists at most one b ∈ Dj such that (a, b) ∈ cij . cij is functional on i if cji is functional on i. When a constraint cij is functional on j, for simplicity, we say cij is functional by making use of the fact that the subscripts of cij are an ordered pair. In this paper, the definition of functional constraints is different from the one in [5, 4] where constraints are functional on each of its variables, leading to the following notion. A constraint cij is bi-functional if cij is functional on both i and j. A bifunctional constraint is called bijective in [2] and simply functional in [4].