Exploiting Single-Cycle Symmetries in Branch-and-Prune algorithms

As a first attempt to exploit symmetries in continuous constraint problems, we focus on permutations of the variables consisting of one single cycle. We propose a procedure that takes advantage of these symmetries by interacting with a Branch-and-Prune algorithm without interfering with it. A key concept in this procedure are the classes of symmetric boxes formed by bisecting a n-dimensional cube at the same point in all dimensions at the same time. We quantify these classes as a function of n. Moreover, we propose a simple algorithm to generate the representatives of all these classes for any number of variables at very high rates. A problem example from the chemical field and a kinematics solver are used to show the performance of the approach in practice. 1 Symmetry in Continuous Constraints Problems Symmetry exploitation in discrete constraint problems has received a great deal of attention lately [6, 4, 5, 11]. On the contrary, symmetries have been largely disregarded in continuous constraint solving, despite the important growth in both theory and applications that this field has recently experienced [12, 1, 9]. Continuous (or numerical) constraint solving is often tackled using Branchand-Prune algorithms [13], which iteratively locate solutions inside an initial domain box, by alternating box subdivision (branching) and box reduction (pruning) steps. Motivated by a molecular conformation problem, in this paper we deal with the most simple type of box symmetry, namely that in which domain variables (i.e., box dimensions) undergo a single-cycle permutation leaving the constraints invariant. This can be seen, thus, as a form of constraint symmetry in the terminology introduced in [3]. We are interested in solving the following general Continuous Constraint Satisfaction Problem (CCSP): Find all points x = (x1, . . . , xn) lying in an initial box of R satisfying the constraints f1(x) ∈ C1 , . . . , fm(x) ∈ Cm, where fi is a function fi : R → R, and Ci is an interval in R. We assume the problem is tackled using a Branch-and-Prune (B&P) algorithm. The only particular feature that we require of this algorithm is that it has to work with boxes in R. We say that a function s : R → R is a point symmetry of the problem if there exists an associated permutation σ ∈ Σm such that fi(x) = fσ(i)(s(x))