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1. On the Budgeted Hausdorff Distance Problem

   Har-Peled, Sariel; Raichel, Benjamin (July 2023, Proceedings of the 35th Canadian Conference on Computational Geometry)

   Pankratov, Denis (Ed.)

   Given a set $$P$$ of $$n$$ points in the plane, and a parameter $$k$$, we present an algorithm, whose running time is $$O(n^{3/2} \sqrt{k}\log^{3/2} n + kn\log^2 n)$$, with high probability, that computes a subset $$Q* \subseteq P$$ of $$k$$ points, that minimizes the Hausdorff distance between the convex-hulls of $Q*$ and $$P$$. This is the first subquadratic algorithm for this problem if $$k$$ is small. 

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2. Convex Polygon Containment: Improving Quadratic to Near Linear Time

   https://doi.org/10.4230/LIPIcs.SoCG.2024.34  

   Chan, Timothy M; Hair, Isaac M (June 2024, Proc. 40th Sympos. Computational Geometry (SoCG))

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0. Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998). 

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3. Online Balanced Allocation of Dynamic Components

   https://doi.org/10.4230/lipics.itcs.2025.81  

   Rajaraman, Rajmohan; Wasim, Omer (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We introduce Online Balanced Allocation of Dynamic Components (OBADC), a problem motivated by the practical challenge of dynamic resource allocation for large-scale distributed applications. In OBADC, we need to allocate a dynamic set of at most k𝓁 vertices (representing processes) in 𝓁 > 0 clusters. We consider an over-provisioned setup in which each cluster can hold at most k(1+ε) vertices, for an arbitrary constant ε > 0. The communication requirements among the vertices are modeled by the notion of a dynamically changing component, which is a subset of vertices that need to be co-located in the same cluster. At each time t, a request r_t of one of the following types arrives: 1) insertion of a vertex v forming a singleton component v at unit cost. 2) merge of (u,v) requiring that the components containing u and v be merged and co-located thereafter. 3) deletion of an existing vertex v at zero cost. Before serving any request, an algorithm can migrate vertices from one cluster to another, at a unit migration cost per vertex. We seek an online algorithm to minimize the total migration cost incurred for an arbitrary request sequence σ = (r_t)_{t > 0}, while simultaneously minimizing the number of clusters utilized. We analyze competitiveness with respect to an optimal clairvoyant offline algorithm with identical (over-provisioned) capacity constraints. We give an O(log k)-competitive algorithm for OBADC, and a matching lower-bound. The number of clusters utilized by our algorithm is always within a (2+ε) factor of the minimum. Furthermore, in a resource augmented setting where the optimal offline algorithm is constrained to capacity k per cluster, our algorithm obtains O(log k) competitiveness and utilizes a number of clusters within (1+ε) factor of the minimum. We also consider OBADC in the context of machine-learned predictions, where for each newly inserted vertex v at time t: i) with probability η > 0, the set of vertices (that exist at time t) in the component of v is revealed and, ii) with probability 1-η, no information is revealed. For OBADC with predictions, we give a O(1)-consistent and O(min(log 1/(η), log k))-robust algorithm. 

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4. Sparse Approximation via Generating Point Sets

   https://doi.org/10.1137/1.9781611974331.ch40  

   Blum, Avrim; Har-Peled, Sariel; Raichel, Benjamin (December 2015, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2016))

   For a set P of n points in the unit ball b ⊆ R d , consider the problem of finding a small subset T ⊆ P such that its convex-hull ε-approximates the convex-hull of the original set. Specifically, the Hausdorff distance between the convex hull of T and the convex hull of P should be at most ε. We present an efficient algorithm to compute such an ε ′ -approximation of size kalg, where ε ′ is a function of ε, and kalg is a function of the minimum size kopt of such an ε-approximation. Surprisingly, there is no dependence on the dimension d in either of the bounds. Furthermore, every point of P can be ε- approximated by a convex-combination of points of T that is O(1/ε2 )-sparse. Our result can be viewed as a method for sparse, convex autoencoding: approximately representing the data in a compact way using sparse combinations of a small subset T of the original data. The new algorithm can be kernelized, and it preserves sparsity in the original input. 

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5. 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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