Guarding galleries where no point sees a small area

Abstract

We prove the following result which is the planar version of a conjecture of Kavraki, Latombe, Motwani, and Raghavan: there is a functionf(h, ɛ) polynomial inh and 1/ɛ such that ifX is a compact planar set of Lebesgue measure 1 withh holes, such that any pointx ∈X sees a part ofX of measure at leastɛ, then there is a setG of at mostf(h, ɛ) points (guards) inX such that any point ofX is seen by at least one point ofG. With a high probability, a setG off(h, ɛ) random points inX (chosen uniformly and independently) has the above property. In the proof (givingf(h, ɛ)≤(2+o(1))1/ɛ log 1/ɛ log2 h) we apply ideas of Kalai and Matousek who proved a weaker boundf(h, ɛ)≤C(h)1/ɛ log 1/ɛ, whereC(h) is a ‘quite fast growing function’ ofh. We improve their bound by showing a stronger result on the so-called VC-dimension of related set systems.