Rigorous Global Filtering Methods with Interval Unions

This paper presents rigorous filtering methods for constraint satisfaction problems based on the interval union arithmetic. Interval unions are finite sets of closed and disjoint intervals that generalize the interval arithmetic. They allow a natural representation of the solution set of interval powers, trigonometric functions and the division by intervals containing zero. We show that interval unions are useful when applied to the forward-backward constraint propagation on directed acyclic graphs (DAGs) and can also replace the interval arithmetic in the Newton operator. Empirical observations support the conclusion that interval unions reduce the search domain even when more expensive state-of-the-art methods fail. Interval unions methods tend to produce a large number of boxes at each iteration. We address this problem by taking a suitable gap-filling strategy. Numerical experiments on constraint satisfaction problems from the COCONUT show the capabilities of the new approach.