有限行无限六边形网格中的最优与准最优定位-支配密度
Optimal and quasi-optimal locating-dominating densities in the infinite hexagonal grid with a finite number of rows
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中文总结 AI 辅助
本文针对有限行无限六边形网格H_k，提出精确指数算法求k≤5的最优解，通过整数线性规划结合H_3最优解得到k≥6的准最优解，偏差不超1.3%，所有解描述极短。
中文摘要 AI 辅助
图G的顶点集S若满足：S是支配集，且对每个不在S中的不同顶点对，它们在S中的邻域互不相同，则称S是定位-支配集。本文研究宽度为k的有限行无限六边形网格（记为H_k，也称为宽度为k的六边形带）中此类集合的最小密度。对每个k≥2，本文给出H_k的最优解或准最优解，其与最优值的偏差不超过1.3%。本文描述了一种针对固定k的精确指数时间算法，通过实现该算法得到了k≤5时的最优解。由于无限网格H_k总能接受周期性最优解，为处理更大的k，本文提出一种整数线性规划，可为每个固定周期找到H_k的最优周期解。该规划为H_7和H_8生成了高质量可行解，再结合H_3的最优解，得到所有k≥6时的准最优解，所有这些解都具有极简短的描述。
英文摘要
A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\geq 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k \leq 5$. As the infinite grid $H_{k}$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_{k}$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\geq 6$. All these solutions admit a very short description.
