Constrained Maximum Weight Paths in Random Geometric Graphs

In this paper, we consider a random geometric graph (RGG)~$G$ formed by~$n$ vertices distributed uniformly in the unit square~$S$ on the plane and equip each edge of~$G$ with an independent weight. We assume that the adjacency distance between vertices is larger than the connectivity threshold and estimate the maximum weight of a path connecting two fixed points~$O_1$ and~$O_2$ in~$S,$ using edges of~$G$ that satisfy length and weight constraints. Our strategy is to first divide the space between~$O_1$ and~$O_2$ into small squares and identify nice squares containing heavy edges. We then use an iterative stitching procedure to connect these heavy edges and estimate the weight of the resulting path~$P\mathrm{.}$ Finally, we invoke a multi-level scaling procedure along with weight segmentation to establish an upper bound for the maximum weight of \emph{any} path between~$O_1$ and~$O_2$ and thereby demonstrate the near-optimality of~$P\mathrm{.}$ We also illustrate our results using examples involving edge weights satisfying power law and exponential decay.