The constraint of difference and total dual integrality

Abstract

One of the most important logic constraints is the constraint of difference. It is imposed on a set of discrete variables requiring that they receive pairwise distinct values. This construct, initially studied in the field of Artificial Intelligence (in particular, Constraint Programming), has numerous applications and important theoretical properties. In the current work, we show that the polytope associated with this constraint is a generalized polymatroid and thus totally dual integral. As a consequence the problem of optimizing a linear function when variables are restricted to take pairwise distinct values belongs to P. Furthermore, we prove that the above problem can be solved by the greedy algorithm in O(|J| · log|J|) steps where J denotes the set indexing the variables (to receive pairwise distinct values). We establish that the dual of the above problem can also be solved in the same number of steps.