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1. Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems

   https://doi.org/10.1287/moor.2021.1181  

   Ghuge, Rohan; Nagarajan, Viswanath (May 2022, Mathematics of Operations Research)

   We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors). 

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2. Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

   https://doi.org/10.1142/S021820251950057X  

   Borthagaray, Juan Pablo; Nochetto, Ricardo H.; Salgado, Abner J. (December 2019, Mathematical Models and Methods in Applied Sciences)

   We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condition. The weight is a power of the distance to the boundary [Formula: see text] of [Formula: see text] that accounts for the singular boundary behavior of the solution for any [Formula: see text]. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension [Formula: see text], which is optimal for [Formula: see text]. 

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3. Proportional Volume Sampling and Approximation Algorithms for A-Optimal Design

   https://doi.org/doi:10.1137/1.9781611975482.84  

   Nikolov, A; Singh, M.; Tantipongpipat, U. (January 2019, Symposium on Discrete Algorithms (SODA))

   We study the A-optimal design problem where we are given vectors υ1, …, υn ∊ ℝd, an integer k ≥ d, and the goal is to select a set S of k vectors that minimizes the trace of (∑i∊Svivi⊺)−1. Traditionally, the problem is an instance of optimal design of experiments in statistics [35] where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick k of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks [22], sparse least squares regression [8], feature selection for k-means clustering [9], and matrix approximation [13, 14, 5]. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for A-optimal design. Given a matrix, proportional volume sampling involves picking a set of columns S of size k with probability proportional to µ(S) times det(∑i∊Svivi⊺) for some measure µ. Our main result is to show the approximability of the A-optimal design problem can be reduced to approximate independence properties of the measure µ. We appeal to hardcore distributions as candidate distributions µ that allow us to obtain improved approximation algorithms for the A-optimal design. Our results include a d-approximation when k = d, an (1 + ∊)-approximation when and -approximation when repetitions of vectors are allowed in the solution. We also consider generalization of the problem for k ≤ d and obtain a k-approximation. We also show that the proportional volume sampling algorithm gives approximation algorithms for other optimal design objectives (such as D-optimal design [36] and generalized ratio objective [27]) matching or improving previous best known results. Interestingly, we show that a similar guarantee cannot be obtained for the E-optimal design problem. We also show that the A-optimal design problem is NP-hard to approximate within a fixed constant when k = d. 

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4. Fast Fréchet Distance Between Curves with Long Edges

   https://doi.org/10.1142/S0218195919500043  

   Gudmundsson, Joachim; Mirzanezhad, Majid; Mohades, Ali; Wenk, Carola (June 2019, International Journal of Computational Geometry & Applications)

   Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve. 

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5. Distributed and Robust Support Vector Machine

   https://doi.org/10.1142/S0218195920500107  

   Liu, Yangwei; Ding, Hu; Huang, Ziyun; Xu, Jinhui (September 2020, International Journal of Computational Geometry & Applications)

   null (Ed.)

   In this paper, we consider the distributed version of Support Vector Machine (SVM) under the coordinator model, where all input data (i.e., points in [Formula: see text] space) of SVM are arbitrarily distributed among [Formula: see text] nodes in some network with a coordinator which can communicate with all nodes. We investigate two variants of this problem, with and without outliers. For distributed SVM without outliers, we prove a lower bound on the communication complexity and give a distributed [Formula: see text]-approximation algorithm to reach this lower bound, where [Formula: see text] is a user specified small constant. For distributed SVM with outliers, we present a [Formula: see text]-approximation algorithm to explicitly remove the influence of outliers. Our algorithm is based on a deterministic distributed top [Formula: see text] selection algorithm with communication complexity of [Formula: see text] in the coordinator model. 

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