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Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an O(n^{4/3}log^{5/3}nlog^{O(1)}log n)-time algorithm for the problem, where n is the total number of all points and halfplanes. This improves the previously best algorithm of n^{10/3}2^{O(log^*n)} time by roughly a quadratic factor. For the special case where all halfplanes are lower ones, our algorithm runs in O(nlog n) time, which improves the previously best algorithm of n^{4/3}2^{O(log^*n)} time and matches an Ω(nlog n) lower bound. Further, our techniques can be extended to solve a star-shaped polygon coverage problem in O(nlog n) time, which in turn leads to an O(nlog n)-time algorithm for computing an instance-optimal ε-kernel of a set of n points in the plane. Agarwal and Har-Peled presented an O(nklog n)-time algorithm for this problem in SoCG 2023, where k is the size of the ε-kernel; they also raised an open question whether the problem can be solved in O(nlog n) time. Our result thus answers the open question affirmatively. 

Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks. 

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem. 

We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (x1, y1), . . . , (xn, yn) ∈ Rd × R where d ∈ {1, 2}, the goal is to segment xi’s into some (arbitrary) number of disjoint pieces P1, . . . , Pk, where each piece Pj is associated with a fixed-degree polynomial fj : Rd → R, to minimize the total loss function λk+􏰄ni=1(yi −f(xi))2, where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f : 􏰇kj=1 Pj → R is the piecewise polynomial function defined as f|Pj = fj. The pieces P1,...,Pk are disjoint intervals of R in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions. Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(nε log1ε), assuming that data is presented in sorted order of xi’s. For bivariate data, we √ present three results: a sub-exponential exact algorithm with running time nO( n); a polynomial-time constant- approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness. 

Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S∩Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on R^2. In this paper, we study RCP search in higher dimensions. We give the first nontrivial RCP data structures for orthogonal, simplex, halfspace, and ball queries in R^d for any constant d. Furthermore, we prove a conditional lower bound for orthogonal RCP search for d≥3.