The minimum constraint removal problem seeks to find the minimum number of constraints, i.e., obstacles, that need to be removed to connect a start to a goal location with a collision-free path. This problem is NP-hard and has been studied in robotics, wireless sensing, and computational geometry. This work contributes to the existing literature by presenting and discussing two results. The first result shows that the minimum constraint removal is NP-hard for simply connected obstacles where each obstacle intersects a constant number of other obstacles. The second result demonstrates that for $n$ simply connected obstacles in the plane, instances of the minimum constraint removal problem with minimum removable obstacles lower than $(n+1)/3$ can be solved in polynomial time. This result is also empirically validated using several instances of randomly sampled axis-parallel rectangles.

翻译：最小约束移除问题旨在寻求从起点到目标点构建无碰撞路径所需移除的最少约束（即障碍物）数量。该问题属于NP难问题，已在机器人学、无线传感和计算几何领域得到广泛研究。本文通过阐述并讨论两项成果为现有文献作出贡献。第一项成果表明：对于每个障碍物仅与常数个其他障碍物相交的单连通障碍物场景，最小约束移除问题仍为NP难问题。第二项成果证明：对于平面上的$n$个单连通障碍物，当最小可移除障碍物数量低于$(n+1)/3$时，最小约束移除问题的实例可在多项式时间内求解。该结论通过多组随机采样轴平行矩形实例进行了实证验证。