AI 中文总结
该研究针对图上马尔可夫移动的多机器人监测问题，分析三种策略模型，证明异构模型下覆盖率最大化的NP难性，给出多种近似算法及不可近似界，并揭示收益递减性质。
AI 中文摘要
我们研究连通图$G=(V,E)$上的随机多机器人监测问题，其中每个机器人根据$G$上的马尔可夫链移动，并监测其当前顶点的闭邻域。$r$个机器人的性能在稳态下通过两个目标评估：平均情况覆盖率（被覆盖顶点的期望数量）和最坏情况覆盖率（所有顶点的最小覆盖概率）。我们考虑三种模型：独立同构策略，即所有机器人共享相同的平稳分布；独立异构策略，即机器人使用不同的平稳分布；以及集中式策略，允许机器人位置之间存在任意相关性。对于异构模型，我们证明即使只有两个机器人，最大化平均覆盖率也是NP难的；复制一种易于计算的最优同构策略可在异构设置下为两个目标函数提供$\boldsymbol{\frac{1}{2}}^{\frac{1}{r}}^r}$近似；此外，除非$\text{P}=\text{NP}$，否则没有多项式时间算法能达到优于$1-\frac{1}{e}$的比率。集中式策略可利用相关性减少冗余。我们建立了一个近似因子层次结构：对于任意正整数$r'\frac{1}{r}$，记$r=hr'+b$且$0\frac{1}{b}<r'$，块协调可为两个目标提供$1-\frac{1}{r'}^h\frac{1}{b}^r$近似。我们还确立了NP难性和紧的$1-\frac{1}{e}$不可近似界。此外，我们证明了关于机器人数量的收益递减性质：非递增比率性质在所有设置下对平均情况目标成立，但对异构最坏情况目标不成立。
英文摘要
We study a stochastic multi-robot monitoring problem on a connected graph $G=(V,E)$, where each robot moves according to a Markov chain on $G$ and monitors the closed neighborhood of its current vertex. The performance of $r$ robots is evaluated in steady state via two objectives: average-case coverage (the expected number of covered vertices) and worst-case coverage (the minimum coverage probability over all vertices). We consider three models: independent homogeneous strategies, where all robots share the same stationary distribution; independent heterogeneous strategies, where robots use different stationary distributions; and centralized strategies, allowing arbitrary correlations between robot locations. For the heterogeneous model, we prove that maximizing average coverage is NP-hard even for two robots, and that replicating an easy-to-compute optimal homogeneous strategy yields a \(\left(1-\left(1-\frac{1}{r}\right)^r\right)\)-approximation for both objective functions in the heterogeneous setting; moreover, no polynomial-time algorithm can achieve a ratio better than \(1-\nicefrac{1}{e}\) unless \(\text{P}=\text{NP}\). Centralized strategies can exploit correlations to reduce redundancy. We develop a hierarchy of approximation factors: for any positive integer \(r'\le r\), writing \(r=hr'+b\) with \(0\le b<r'\), block coordination yields a \(1-\left(1-\frac{r'}{r}\right)^h\left(1-\frac{b}{r}\right)\) approximation for both objectives. We also establish NP-hardness and a tight \(1-\nicefrac{1}{e}\) inapproximability bound. Moreover, we prove diminishing-returns properties with respect to the number of robots: a non-increasing-ratio property holds for the average-case objective in all settings, but not for the heterogeneous worst-case objective.
