Minimum Ply Covering of Points with Unit Squares
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Given a set P of points and a set U of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of squares that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. [Comput. Geom., 94:101712, 2020] showed that determining the minimum ply cover number for a set of points by a set of axis-parallel unit squares is NP-hard, and gave a polynomial-time 2-approximation algorithm for instances in which the minimum ply cover number is constant. The question of whether there exists a polynomial-time approximation algorithm remained open when the minimum ply cover number is ω(1). We settle this open question and present a polynomial-time (8+ε)-approximation algorithm for the general problem, for every fixed ε>0.
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