Efficient Algorithms for the Bottleneck Path Problem in Geometric Graphs

2026-08-04

AI Summary

This work addresses the bottleneck path problem in two classes of geometric graphs: directional antenna graphs in the plane with a lower-bounded antenna angle, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain. For such graphs equipped with Euclidean edge weights, the authors present a near-linear-time algorithm to decide whether a given threshold subgraph contains a feasible path. By integrating this decision procedure with parametric search and randomization techniques, they achieve—for the first time—an expected-time bottleneck path algorithm with complexity \(O^*(n^{8/7})\), where sublinear factors are absorbed into the \(O^*\) notation. The approach naturally extends to hop-constrained variants and demonstrates efficient performance in both geometric settings, providing a unified solution for both standard and bounded-hop scenarios.

Abstract

We present efficient algorithms for the bottleneck path problem in two geometric settings that arise naturally in applications: directional-antenna graphs in the plane with antenna angles bounded from below by a constant, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain, both with Euclidean distances as edge weights. We provide near-linear algorithms for the corresponding decision problems, namely, determining whether the subgraph obtained by retaining all edges with weight at most some threshold ${\bf bn}$ contains a path from $s$ to $t$. We then use the decision procedures to obtain algorithms for the bottleneck path problem that run in $O^*(n^{8/7})$ randomized expected time, where $n$ is the input size and the $O^*(\cdot)$ notation hides subpolynomial factors. Within the same performance bounds, we can also solve the bounded-hop version, in which we only consider $s$-$t$ paths with at most $k$ edges, for a given integer $k < n$.

Problem

• bottleneck path problem
• geometric graphs
• directional-antenna graphs
• visibility graphs
• bounded-hop paths

Innovation

• bottleneck path
• geometric graphs
• near-linear algorithms
• randomized algorithms
• bounded-hop paths