Linear 0-1 Inequalities and Extended Clauses

Extended clauses are the basic formulas of the 0-1 constraint solver for the constraint logic programming language CLP(PB). We present a method for transforming an arbitrary linear 0-1 inequality into a set of extended clauses, such that the solution space remains invariant. After applying well-known linearization techniques on non-linear 0-1 constraints followed by the presented transformation method, we are able to handle arbitrary 0-1 constraints in CLP(PB). The transformation method presented relies on cutting planes techniques known from 0-1 integer programming. We develop specialized redundancy criteria and so produce the minimal number of extended clauses needed for preserving equivalence. The method is enhanced by using a compact representation of linear 0-1 inequalities and extended clauses. Unit resolution for classical clauses is generalized to pseudo-Boolean unit resolution for arbitrary linear 0-1 inequalities. We extend the transformation method to constrained transformation when the inequality to be transformed is part of a larger set of linear 0-1 inequalities. Furthermore the method can be used to obtain all strongest extended cover inequalities of a knapsack inequality.