Helly-Type Theorems for Approximate Covering

Let <i>F</i> ∪ {<i>U</i>} be a collection of convex sets in R^d such that <i>F</i> covers <i>U</i>. We show that if the elements of <i>F</i> and <i>U</i> have comparable size, in the sense that each contains a ball of radius <i>r</i> and is contained in a ball of radius <i>R</i> for some fixed <i>r</i> and <i>R</i>, then for any ε > 0 there exists <i>H</i>_ε ⊂ <i>F</i>, whose size |<i>H</i>_ε| is polynomial in 1/ε and independent of |<i>F</i>|, that covers <i>U</i> except for a volume of at most ε. The size of the smallest such subset depends on the geometry of the elements of <i>F</i>; specifically, we prove that it is <i>O</i>(1/ε) when <i>F</i> consists of axis-parallel unit squares in the plane and <i>Õ</i>(ε^{1--<i>d</i>/2}) when <i>F</i> consists of unit balls in R^d (here, <i>Õ</i>(<i>n</i>) means <i>O</i>(<i>n</i> log <i>n</i>) for some constant), and that these bounds are, in the worst-case, tight up to the logarithmic factors. We extend these results to surface-to-surface visibility in 3 dimensions: if a collection <i>F</i> of disjoint unit balls occludes visibility between two balls then a subset of <i>F</i> of size <i>Õ</i>(ε^--7/2) blocks visibility along all but a set of lines of measure ε. Finally, for each of the above situations we give an algorithm that takes <i>F</i> and <i>U</i> as input and outputs in time <i>O</i>(|<i>F</i>|*|<i>H</i>_ε|) either a point in <i>U</i> not covered by <i>F</i> or a subset <i>H</i>_ε covering <i>U</i> up to a measure ε, with |<i>H</i>_ε| satisfying the above bounds.