An Analysis of Slow Convergence in Interval Propagation

Abstract

When performing interval propagation on integer variables with a large range, slow-convergence phenomena are often observed: it becomes difficult to reach the fixpoint of the propagation. This problem is practically important, as it hinders the use of propagation techniques for problems with large numerical ranges, and notably problems arising in program verification. A number of attempts to cope with this issue have been investigated, yet all of the proposed techniques only guarantee a fast convergence on specific instances. An important question is therefore whether slow convergence is intrinsic to propagation methods, or whether an improved propagation algorithm may exist that would avoid this problem. This paper proposes the first analysis of the slow convergence problem under the light of complexity results. It answers the question, by a negative result: if we allow propagators that are general enough, computing the fixpoint of constraint propagation is shown to be intractable. Slow convergence is therefore unavoidable unless P=NP. The result holds for the propagators of a basic class of constraints.

An unabridged version of this paper including the missing proofs is available as a Microsoft Research Report.