A Covering Problem for Hypercubes

We introduce a new NP-complete problem asking if a “query” hypercube is (not) covered by a set of other “evidence” hypercubes. This comes down to a form of constraint reasoning asking for the satisfiability of a CNF formula where the logical atoms are inequalities over single variables, with possibly infinite variable domains. We empirically investigate the location of the phase transition regions in two random distributions of problem instances. We introduce a solution method that iteratively constructs a representation of the non-covered part of the query cube. In particular, the method is not based on backtracking. Our experiments show that the method is, in a significant range of instances, superior to the backtracking method that results from translation to SAT, and application of a stateof-the-art DP-based SAT solver. This paper is an extended abstract. More details can be found in the long version of the paper [Hoffmann and Kupferschmid, 2005]. We introduce a new NP-complete problem asking if there is a point in a given n-dimensional “query” hypercube that is not covered by – contained in the union of – a set of other ndimensional “evidence” hypercubes. An n-dimensional hypercube is a cross product of n intervals. Intervals in our context are defined as statements of the form l < [≤] x < [≤] u where “x” is a variable, “l” and “u” are members of x’s domain, and “<” is a total order defined over this domain.1 Definition 1 Let QCOVER denote the following problem: Given an n-dimensional hypercube Q, and a set E of ndimensional hypercubes, is there a point in Q that is not contained in ⋃ E∈E E? Covering problems of this kind arise, e.g., in the context of regression planning with numeric state variables [Koehler, 1998]. More generally, QCOVER is a form of constraint reasoning asking for the satisfiability of a CNF formula where the logical atoms are inequalities over single variables, with possibly infinite variable domains. The correspondence is the By square parentheses “symb1 [symb2]” we denote alternative possibilities, i.e. that symb2 can be substituted for symb1. following. A QCOVER instance is a constraint problem with n variables x. The query hypercube specifies a region inside which the solution must lie, the evidence cubes specify regions inside which the solution must not lie. A hypercube corresponds to a conjunction of inequalities of the form c < [≤, >,≥] x. So the complement of a hypercube (of an evidence hypercube) corresponds to a disjunction of such inequalities, and the overall problem is a conjunction of disjunctive constraints. Vice versa, any conjunction of such disjunctive constraints can be expressed as hypercubes (if a disjunctive constraint does not mention a variable x, then the interval in dimension d is the whole variable domain). Proposition 1 QCOVER is NP-complete. We empirically explore two random distributions of QCOVER instances. The first one, which we call Random QCOVER, chooses the end points for all intervals uniformly from a set of m possible values. The second one, which we call Random 3-QCOVER, is similar to the fixed clause-length model for generating random 3SAT instances [Mitchell et al., 1992]. It always selects the query cube to be the cross-product of the (whole) variable domains, and, in the evidence cubes, assigns the whole variable domains to all but 3 randomly chosen dimensions. For both distributions, we investigate the location of the phase transition regions. As it turns out, Random 3-QCOVER shows a typical phase transition behaviour while Random QCOVER shows no such behaviour, see Figure 1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1