A Partition-Based Heuristic for Translational Box Covering

Geometric covering problems are found in application domains such as sensor coverage and repairing of materials. This paper considers covering problems in which a target shape to be covered is specified and a set of covering shapes is supplied. The goal is to find translational positions of the covering shapes that allow them to collectively cover the target. All the shapes are orthotopes (convex boxes). The approach provides the best known results for translational, two-dimensional orthotope covering and the first results in three and four dimensions. Experiments suggest why the strategy succeeds in certain regions of the problem domain in spite of the NP-hardness of translational orthotope covering. This leads to a novel, dimension-independent measure that significantly improves the speed of the method. Orthotopes can form enclosures for nonconvex shapes, so progress for orthotope covering is beneficial in a more general context. Key–Words: Geometric covering, orthotopes, Lagrangian relaxation, simulated annealing