Solving optimization problems leads to elegant and practical solutions in a wide variety of real-world applications. In many of those real-world applications, some of the information required to specify the relevant optimization problem is noisy, uncertain, and expensive to obtain. In this work, we study how much of that information needs to be queried in order to obtain an approximately optimal solution to the relevant problem. In particular, we focus on the shortest path problem in graphs with dynamic edge costs. We adopt the $\textit{first passage percolation}$ model from probability theory wherein a graph $G'$ is derived from a weighted base graph $G$ by multiplying each edge weight by an independently chosen random number in $[1, \rho]$. Mathematicians have studied this model extensively when $G$ is a $d$-dimensional grid graph, but the behavior of shortest paths in this model is still poorly understood in general graphs. We make progress in this direction for a class of graphs that resemble real-world road networks. Specifically, we prove that if $G$ has a constant continuous doubling dimension, then for a given $s-t$ pair, we only need to probe the weights on $((\rho \log n )/ \epsilon)^{O(1)}$ edges in $G'$ in order to obtain a $(1 + \epsilon)$-approximation to the $s-t$ distance in $G'$. We also generalize the result to a correlated setting and demonstrate experimentally that probing improves accuracy in estimating $s-t$ distances.

翻译：解决优化问题在众多实际应用场景中催生了优雅且实用的方案。然而，在许多此类应用中，定义相关优化问题所需的信息往往带有噪声、不确定性，且获取成本高昂。本研究探讨了为获得相关问题的近似最优解而需查询的信息量。具体而言，我们聚焦于具有动态边成本的图中的最短路径问题。我们采用概率论中的"首达逾渗"模型，其中图G'通过将加权基图G的每条边权乘以独立选自区间[1,ρ]的随机数而得到。数学家已对G为d维网格图的情形进行了广泛研究，但一般图中该模型的最短路径行为仍知之甚少。我们在类似现实道路网络的图类上取得进展：若G具有常数连续倍维数，则对于给定的s-t对，仅需探测G'中((ρ log n)/ε)^O(1)条边的权值，即可获得G'中s-t距离的(1+ε)近似解。此外，我们将结果推广至相关设置，并通过实验证明探测可提高s-t距离估计的准确性。