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

Abstract

In the $d$-dimensional cow-path problem, a cow living in ${\mathbb{R}}^d$ must locate a $(d-1)$-dimensional hyperplane $H$ whose location is unknown. The only way that the cow can find $H$ is to roam ${\mathbb{R}}^d$ until it intersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate a hyperplane $H$ whose distance from the origin was $r\ge 1$, then the cow is said to achieve competitive ratio $s/r$. It is a classic result that, in ${\mathbb{R}}^2$, the optimal (deterministic) competitive ratio is $9$. In ${\mathbb{R}}^3$, the optimal competitive ratio is known to be at most $\approx 13.811$. 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 $\Omega (d)$, leaving a gap of roughly $\sqrt{d}$. In this note, we achieve a stronger lower bound of $\tilde{\Omega }(d^{3/2})$.