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1. Clustering with Neighborhoods

   https://doi.org/10.4230/LIPIcs.ISAAC.2021.6  

   Huang, Hongyao ; Klimenko, Georgiy ; Raichel, Benjamin ( November 2021 , International Symposium on Algorithms and Computation (ISAAC))

   Ahn, Hee-Kap ; Sadakane, Kunihiko (Ed.)

   In the standard planar k-center clustering problem, one is given a set P of n points in the plane, and the goal is to select k center points, so as to minimize the maximum distance over points in P to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the k-center problem to allow the covered objects to be a set of general disjoint convex objects C rather than just a point set P. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if r_opt is the optimal radius for k centers, then in n^O(1/ε²) time we can produce a set of (1+ε)k centers with radius ≤ r_opt. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping k as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P = NP, even when C is a set of line segments. When C is a set of unit disks we show the problem is hard to approximate within a factor of (√{13}-√3)(2-√3) ≈ 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+2√3)≈ 8.46 approximation algorithm. Additionally, for unit disks we give an O(n log k)+(k/ε)^O(k) time (1+ε)-approximation to the optimal radius, that is, an FPTAS for constant k whose running time depends only linearly on n. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(n log n) time. 

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2. Which Lp norm is the fairest? Approximations for fair facility location across all "p"

   https://doi.org/10.1145/3580507.3597664  

   Gupta, Swati ; Moondra, Jai ; Singh, Mohit ( July 2023 , EC '23: Proceedings of the 24th ACM Conference on Economics and Computation)

   Given a set of facilities and clients, and costs to open facilities, the classic facility location problem seeks to open a set of facilities and assign each client to one open facility to minimize the cost of opening the chosen facilities and the total distance of the clients to their assigned open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms. In this work, we consider a fair version of the problem where we are given 𝑟 clients groups that partition the set of clients, and the distance of a given group is defined as the average distance of clients in the group to their respective open facilities. The objective is to minimize the Minkowski 𝑝-norm of vector of group distances, to penalize high access costs to open facilities across 𝑟 groups of clients. This generalizes classic facility location (𝑝 = 1) and the minimization of the maximum group distance (𝑝 = ∞). However, in practice, fairness criteria may not be explicit or even known to a decision maker, and it is often unclear how to select a specific "𝑝" to model the cost of unfairness. To get around this, we study the notion of solution portfolios where for a fixed problem instance, we seek a small portfolio of solutions such that for any Minkowski norm 𝑝, one of these solutions is an 𝑂(1)-approximation. Using the geometric relationship between various 𝑝-norms, we show the existence of a portfolio of cardinality 𝑂(log 𝑟), and a lower bound of (\sqrt{log r}). There may not be common structure across different solutions in this portfolio, which can make planning difficult if the notion of fairness changes over time or if the budget to open facilities is disbursed over time. For example, small changes in 𝑝 could lead to a completely different set of open facilities in the portfolio. Inspired by this, we introduce the notion of refinement, which is a family of solutions for each 𝑝-norm satisfying a combinatorial property. This property requires that (1) the set of facilities open for a higher 𝑝-norm must be a subset of the facilities open for a lower 𝑝-norm, and (2) all clients assigned to an open facility for a lower 𝑝-norm must be assigned to the same open facility for any higher 𝑝-norm. A refinement is 𝛼-approximate if the solution for each 𝑝-norm problem is an 𝛼-approximation for it. We show that it is sufficient to consider only 𝑂(log 𝑟) norms instead of all 𝑝-norms, 𝑝 ∈ [1, ∞] to construct refinements. A natural greedy algorithm for the problem gives a poly(𝑟)-approximate refinement, which we improve to poly(r^1/\sqrt{log 𝑟})-approximate using a recursive algorithm. We improve this ratio to 𝑂(log 𝑟) for the special case of tree metric for uniform facility open cost. Our recursive algorithm extends to other settings, including to a hierarchical facility location problem that models facility location problems at several levels, such as public works departments and schools. A full version of this paper can be found at https://arxiv.org/abs/2211.14873. 

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3. Improved Algorithms for Distance Selection and Related Problems

   Wang, Haitao ; Zhao, Yiming ( August 2023 , Proceedings of the 31st Annual European Symposium on Algorithms (ESA 2023))

   In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set P of n points in the plane and an integer 1 ≤ k ≤ binom(n,2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of P. The previously best deterministic algorithm solves the problem in O(n^{4/3} log² n) time [Katz and Sharir, 1997]. In this paper, we improve their algorithm to O(n^{4/3} log n) time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fréchet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, 2015] by a factor of roughly log²(m+n) (resp., (m+n)^ε), where m and n are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future. 

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4. Reverse shortest path problem for unit-disk graphs

   Wang, Haitao ; Zhao, Yiming ( April 2023 , Journal of computational geometry)

   Given a set P of n points in the plane, the unit-disk graph Gr(P) with respect to a parameter r is an undirected graph whose vertex set is P such that an edge connects two points p, q in P if the Euclidean distance between p and q is at most r (the weight of the edge is 1 in the unweighted case and is the distance between p and q in the weighted case). Given a value \lambda>0 and two points s and t of P, we consider the following reverse shortest path problem: computing the smallest r such that the shortest path length between s and t in Gr(P) is at most \lambda. In this paper, we present an algorithm of O(\lfloor \lambda \rfloor \cdot n log n) time and another algorithm of O(n^{5/4} log^{7/4} n) time for the unweighted case, as well as an O(n^{5/4} log^{5/2} n) time algorithm for the weighted case. We also consider the L1 version of the problem where the distance of two points is measured by the L1 metric; we solve the problem in O(n log^3 n) time for both the unweighted and weighted cases. 

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5. Dynamic Convex Hulls Under Window-Sliding Updates

   Wang, Haitao ( August 2023 , Proceedings of the 18th Algorithms and Data Structures Symposium (WADS 2023))

   We consider the problem of dynamically maintaining the convex hull of a set S of points in the plane under the following special sequence of insertions and deletions (called window-sliding updates): insert a point to the right of all points of S and delete the leftmost point of S. We propose an O(|S|)-space data structure that can handle each update in O(1) amortized time, such that all standard binary-search-based queries on the convex hull of S can be answered in 𝑂(log |S|) time, and the convex hull itself can be output in time linear in its size. 

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