A Nearly Tight Lower Bound for the $d$-Dimensional Cow-Path Problem

Abstract

In the d-dimensional cow-path problem, a cow living in R d must locate a (d − 1)-dimensional hyperplane H whose location is unknown. The only way that the cow can find H is to roam R d until it intersects H . If the cow travels a total distance s to locate a hyperplane H whose distance from the origin was r ≥ 1, then the cow is said to achieve competitive ratio s/r.It is a classic result that, in R 2 , the optimal (deterministic) competitive ratio is 9. In R 3 , the optimal competitive ratio is known to be at most ≈ 13.811. But in higher dimensions, the asymptotic relationship between d and the optimal competitive ratio remains an open question. The best upper and lower bounds, due to Antoniadis et al., are O(d 3/2 ) and Ω(d), leaving a gap of roughly √ d. In this note, we achieve a stronger lower bound of Ω(d 3/2 ).