Minimum-Membership Geometric Set Cover, Revisited
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We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set S of points and a set ℛ of geometric objects, and the goal is to find a subset ℛ^*⊆ℛ to cover all points in S such that the membership of S with respect to ℛ^*, denoted by 𝗆𝖾𝗆𝖻(S,ℛ^*), is minimized, where 𝗆𝖾𝗆𝖻(S,ℛ^*)=max_p∈ S|{R∈ℛ^*: p∈ R}|. We achieve the following two main results. * We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time n^O(𝗈𝗉𝗍) where 𝗈𝗉𝗍 is the optimum of the problem (i.e., the minimum membership). * We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of o(log n) in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find ℛ^*⊆ℛ to cover S such that the ply of ℛ^* is minimized, where the ply is defined as the maximum number of objects in ℛ^* which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in O(n^12) time. We give a significantly simpler constant-approximation algorithm with near-linear running time.
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