First Passage Percolation with Queried Hints

Optimization problems are ubiquitous throughout the modern world. In many of these applications, the input is inherently noisy and it is expensive to probe all of the noise in the input before solving the relevant optimization problem. In this work, we study how much of that noise needs to be queried in order to obtain an approximately optimal solution to the relevant problem. We focus on the shortest path problem in graphs, where one may think of the noise as coming from real-time traffic. We consider the following model: start with a weighted base graph $G$ and multiply each edge weight by an independently chosen, uniformly random number in $[1,2]$ to obtain a random graph $G'$. This model is called \emph{first passage percolation}. 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 resembles real-world road networks. Specifically, we prove that if the geometric realization of $G$ has constant doubling dimension, then for a given $s-t$ pair, we only need to probe the weights on $((\log n) / \epsilon)^{O(1)}$ edges in $G'$ in order to obtain a $(1 + \epsilon)$-approximation to the $s-t$ distance in $G'$. We also demonstrate experimentally that this result is pessimistic -- one can even obtain a short path in $G'$ with a small number of probes to $G'$.

• Data Mining and Modeling

• Algorithms and Theory