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

TL;DR

This paper establishes a nearly tight lower bound of approximately d^{3/2} for the competitive ratio in the d-dimensional cow-path problem, narrowing the gap between known bounds in higher dimensions.

Contribution

It provides a significantly improved lower bound of d^{3/2} for the competitive ratio, advancing understanding of the problem's complexity in higher dimensions.

Findings

Lower bound of d^{3/2} for the competitive ratio in d-dimensional cow-path problem.

The new bound narrows the gap between existing upper and lower bounds.

The result applies to deterministic algorithms in the cow-path search 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 $\tilde{\Omega}(d^{3/2})$.