For a given polygonal region $P$, the Lawn Mowing Problem (LMP) asks for a shortest tour $T$ that gets within Euclidean distance 1/2 of every point in $P$; this is equivalent to computing a shortest tour for a unit-diameter cutter $C$ that covers all of $P$. As a generalization of the Traveling Salesman Problem, the LMP is NP-hard; unlike the discrete TSP, however, the LMP has defied efforts to achieve exact solutions, due to its combination of combinatorial complexity with continuous geometry. We provide a number of new contributions that provide insights into the involved difficulties, as well as positive results that enable both theoretical and practical progress. (1) We show that the LMP is algebraically hard: it is not solvable by radicals over the field of rationals, even for the simple case in which $P$ is a $2\times 2$ square. This implies that it is impossible to compute exact optimal solutions under models of computation that rely on elementary arithmetic operations and the extraction of $k$th roots, and explains the perceived practical difficulty. (2) We exploit this algebraic analysis for the natural class of polygons with axis-parallel edges and integer vertices (i.e., polyominoes), highlighting the relevance of turn-cost minimization for Lawn Mowing tours, and leading to a general construction method for feasible tours. (3) We show that this construction method achieves theoretical worst-case guarantees that improve previous approximation factors for polyominoes. (4) We demonstrate the practical usefulness \emph{beyond polyominoes} by performing an extensive practical study on a spectrum of more general benchmark polygons: We obtain solutions that are better than the previous best values by Fekete et al., for instance sizes up to $20$ times larger.

翻译：对于给定的多边形区域$P$，割草问题（LMP）要求找到一条最短路径$T$，使其与$P$内每一点的欧氏距离不超过1/2；这等价于为直径为1的圆形刀具$C$计算一条能覆盖整个$P$的最短路径。作为旅行商问题的推广，LMP是NP难的；然而，与离散TSP不同，LMP因组合复杂性与连续几何的结合，一直未能实现精确求解。我们提供了一系列新贡献，既揭示了相关困难的本质，也为理论和实践进展提供了积极结果。（1）我们证明LMP在代数上是困难的：即使对于$P$为$2\times 2$正方形这一简单情形，它也无法在有理数域上通过根式求解。这意味着在依赖基本算术运算和$k$次根提取的计算模型下，无法计算精确最优解，从而解释了实践中观察到的困难。（2）我们利用这一代数分析方法处理一类自然的多边形——边平行于坐标轴且顶点为整数的多边形（即多联骨牌），强调了割草路径中转向代价最小化的相关性，并由此提出一种通用可行路径构造方法。（3）我们证明该构造方法在多联骨牌问题上具有理论最坏情况保证，改进了之前的近似比。（4）我们通过在一系列更通用的基准多边形上进行广泛的实践研究，展示了该方法*超越多联骨牌*的实际效用：我们获得的解优于Fekete等人先前的最佳结果，且实例规模达到了之前的$20$倍以上。