A nearly tight lower bound for the d-dimensional cow-path problem

Abstract

In the d-dimensional cow-path problem, a cow living in Rd must locate a (d−1)-dimensional hyperplane H whose location is unknown. The only way that the cow can find H is to roam Rd 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 R2, the optimal (deterministic) competitive ratio is 9. In R3, 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(d3/2) and Ω(d), leaving a gap of roughly d. In this note, we achieve a stronger lower bound of Ω˜(d3/2).