Hardness of Approximation for Euclidean k-Median

The Euclidean k-median problem is defined in the following manner: given a set ? of n points in d-dimensional Euclidean space ?^d, and an integer k, find a set C ? ?^d of k points (called centers) such that the cost function ?(C,?) ? ?_{x ? ?} min_{c ? C} ?x-c?? is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015]. Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output ? k centers (for constant ? > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any ? < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of ? < 1.28, again assuming UGC

On computing the diameter of a point set in high dimensional Euclidean space

We consider the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space under Euclidean distance function. We describe an algorithm that in time $O(dnlog n +n^{2})$ finds with high probability an arbitrarily close approximation of the diameter. For large values of $d$ the complexity bound of our algorithm is a substantial improvement over the complexity bounds of previously known exact algorithms. Computing and approximating the diameter are fundamental primitives in high dimensional computational geometry and find practical application, for example, in clustering operations for image databases

Mean field and corrections for the Euclidean Minimum Matching problem

Consider the length $L_{MM}^E$ of the minimum matching of N points in d-dimensional Euclidean space. Using numerical simulations and the finite size scaling law $= \beta_{MM}^E(d) N^{1-1/d}(1+A/N+... )$, we obtain precise estimates of $\beta_{MM}^E(d)$ for $2 \le d \le 10$. We then consider the approximation where distance correlations are neglected. This model is solvable and gives at $d \ge 2$ an excellent ``random link'' approximation to $\beta_{MM}^E(d)$. Incorporation of three-link correlations further improves the accuracy, leading to a relative error of 0.4% at d=2 and 3. Finally, the large d behavior of this expansion in link correlations is discussed.Comment: source and one figure. Submitted to PR

Approximating the Geometric Edit Distance

Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a geometric space. Therefore, the geometric edit distance (GED) has been studied. In this paper, we describe the first strictly sublinear approximate near-linear time algorithm for computing the GED of two point sequences in constant dimensional Euclidean space. Specifically, we present a randomized O(n log^2n) time O(sqrt n)-approximation algorithm. Then, we generalize our result to give a randomized alpha-approximation algorithm for any alpha in [1, sqrt n], running in time O~(n^2/alpha^2). Both algorithms are Monte Carlo and return approximately optimal solutions with high probability