Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weighted, axis-aligned rectangles such that the total weight of the rectangles in the cover is minimized. Already the unit-weight case is known to be NP-hard and the general problem has, to the best of our knowledge, not been studied experimentally before. We show a new basic property of optimal solutions of the weighted problem. This allows us to speed up existing algorithms for the unit-weight case, obtain an improved ILP formulation for both the weighted and unweighted problem, and develop several approximation algorithms and heuristics for the weighted case. All our algorithms are evaluated in a large experimental study on 186 837 polygons combined with six cost functions, which provides evidence that our algorithms are both fast and yield close-to-optimal solutions in practice.

翻译：用一组简单形状表示多边形在不同应用场景中具有广泛用途。本文研究用一组带权重的轴对齐矩形覆盖含孔洞平直多边形内部区域的问题，目标是最小化覆盖中所有矩形的总权重。已知单位权重情形已被证明是NP难的，而据我们所知，该一般性问题此前尚未得到实验研究。我们揭示了带权问题最优解的一个新基本性质。基于此性质，我们能够加速现有单位权重问题的算法，获得改进的加权与未加权问题的整数线性规划（ILP）公式，并为加权情形开发多种近似算法与启发式方法。所有算法均在186,837个多边形与六种成本函数的大规模实验研究中进行评估，实验结果表明我们的算法兼具高效性，且在实际中能获得接近最优的解。