Escaping Off-Line Searchers and a Discrete Isoperimetric Theorem

Abstract

Given a set of searchers in the grid, whose search paths are known in advance, can a target that moves at the same speed as the searchers escape detection indefinitely? We study the number of searchers against which the target can still escape. This is less than n in an n×n grid, since a row of searchers can sweep the allowed region.

In an alternating move model where at each time first all searchers move and then the target moves, we show that a target can always escape $\lfloor{1\over2}n\rfloor$ searchers and there is a strategy for $\lfloor{1\over2}n\rfloor+1$ searchers to catch the target. This improves a recent bound $\Omega(\sqrt{n})$ [5] in the simultaneous move model. We also prove similar bounds for the continuous analogue, as well as for searchers and targets moving with different speeds. In the proof, we use a new isoperimetric theorem for subsets of the n×n grid, which is of independent interest.