Autonomous ride-hailing platforms must strategically position idle robotaxis to minimize the wait times of prospective riders. We formalize this as the \emph{robotaxi placement problem} ($k$-RP). Given a finite metric space and a demand distribution over its points, the goal is to position $k$ robotaxis to minimize the expected total distance in a perfect matching between the robotaxis and $k$ random riders. We present several theoretical results for this stochastic optimization problem. First, we observe that sampling robotaxi locations independently according to the demand distribution yields a randomized $2$-approximation algorithm. Second, we present an explicit inapproximability bound via a novel gap-preserving reduction from the maximum coverage problem. Furthermore, while it is not even clear whether the exact expected cost of a placement can be computed efficiently on general metrics, we design an exact polynomial-time dynamic programming algorithm for $k$-RP in tree metrics by decoupling the stochastic matching dependencies. Finally, empirical evaluations on real-world ride-hailing data reveal that a variance-reduced random placement strategy is highly effective in practice, yielding expected wait times that are very close to those obtained by computationally heavy exact algorithms for the uniform capacitated $k$-median problem.

翻译：自主网约车平台必须策略性地部署空闲机器人出租车，以最小化潜在乘客的等待时间。我们将此形式化为"机器人出租车布局问题"（$k$-RP）。给定一个有限度量空间及其点上的需求分布，目标是部署$k$辆机器人出租车，使得在机器人出租车与$k$个随机乘客的完美匹配中，预期总距离最小化。我们针对这一随机优化问题提出了若干理论结果。首先，我们观察到根据需求分布独立采样机器人出租车位置，可得到一个随机化的$2$-近似算法。其次，通过一种新颖的保间隙归约（从最大覆盖问题出发），我们给出了一个显式的不可近似性界。此外，虽然在一般度量空间中，甚至无法明确判定某个布局的精确预期成本能否高效计算，但我们针对树度量空间中的$k$-RP问题，通过解耦随机匹配依赖关系，设计了一种精确的多项式时间动态规划算法。最后，基于真实网约车数据的实证评估表明，方差缩减的随机布局策略在实践中非常有效，其产生的预期等待时间与针对均匀容量限制的$k$-中值问题的高计算复杂度精确算法所获得的结果非常接近。