To catch a falling robber

TL;DR

This paper analyzes a combinatorial pursuit game on subsets of an n-set, establishing bounds on the number of cops needed to guarantee catching the robber at the middle level, improving understanding of pursuit-evasion dynamics.

Contribution

The paper provides an upper bound on the number of cops needed, within a logarithmic factor of the previously known lower bounds, for the subset-based Cops-and-Robber game.

Findings

Upper bound within O(ln n) factor of lower bound

Established bounds for both even and odd n

Enhanced understanding of pursuit strategies on subset lattices

Abstract

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one level by gaining an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Aaron Hill posed the problem and provided a lower bound of $2^{n/2}$ for even $n$ and $\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor}$ for odd $n$. We prove an upper bound (for all $n$) that is within a factor of $O(\ln n)$ times this lower bound.