On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k
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We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in O(n^3/2) time. - We prove a lower bound of Ω(n^4/3-δ) for rectilinear discrete 3-center in 4D, for any constant δ>0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in O(n^8/5) time. We also prove a lower bound of Ω(n^3/2-δ) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is O(n^7/4). - We prove a lower bound of Ω(n^2-δ) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of O(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^k-δ) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω=2. - We also prove an Ω(n^2-δ) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D. This matches the straightforward near-quadratic upper bound.
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