Related papers: A Fast Approximation Scheme for Low-Dimensional $k…
The construction of $r$-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate $r$-nets with respect to Euclidean…
We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J. ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset…
We study the design of efficient approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in high dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction.…
This paper considers the well-studied algorithmic regime of designing a $(1+\epsilon)$-approximation algorithm for a $k$-clustering problem that runs in time $f(k,\epsilon)poly(n)$ (sometimes called an efficient parameterized approximation…
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…
We study the k nearest neighbors problem in the plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals and with respect to a general family of distance…
We present a new approach to approximate nearest-neighbor queries in fixed dimension under a variety of non-Euclidean distances. We are given a set $S$ of $n$ points in $\mathbb{R}^d$, an approximation parameter $\varepsilon > 0$, and a…
The fuzzy $K$-means problem is a generalization of the classical $K$-means problem to soft clusterings, i.e. clusterings where each points belongs to each cluster to some degree. Although popular in practice, prior to this work the fuzzy…
Similarity search based on a distance function in metric spaces is a fundamental problem for many applications. Queries for similar objects lead to the well-known machine learning task of nearest-neighbours identification. Many data…
We consider the task of fitting low-dimensional embeddings to high-dimensional data. In particular, we study the $k$-Euclidean Metric Violation problem ($\textsf{$k$-EMV}$), where the input is $D \in \mathbb{R}^{\binom{n}{2}}_{\geq 0}$ and…
An important goal in algorithm design is determining the best running time for solving a problem (approximately). For some problems, we know the optimal running time, assuming certain conditional lower bounds. In this work, we study the…
This paper develops an efficient algorithm for computing the Euclidean projection onto the top-k-sum constraint, a key operation in financial risk management and matrix optimization problems. Existing projection methods rely on sorting and…
We give a new general approach for designing exact exponential-time algorithms for subset problems. In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the…
Knapsack is one of the most fundamental problems in theoretical computer science. In the $(1 - \epsilon)$-approximation setting, although there is a fine-grained lower bound of $(n + 1 / \epsilon) ^ {2 - o(1)}$ based on the $(\min,…
For very large values of $k$, we consider methods for fast $k$-means clustering of massive datasets with $10^7\sim10^9$ points in high-dimensions ($d\geq100$). All current practical methods for this problem have runtimes at least…
The approximate nearest neighbor problem ($\epsilon$-ANN) in high dimensional Euclidean space has been mainly addressed by Locality Sensitive Hashing (LSH), which has polynomial dependence in the dimension, sublinear query time, but…
The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total…
Let $P$ be a set of points in some metric space. The approximate furthest neighbor problem is, given a second point set $C,$ to find a point $p \in P$ that is a $(1+\epsilon)$ approximate furthest neighbor from $C.$ The dynamic version is…
We present a new local-search algorithm for the $k$-median clustering problem. We show that local optima for this algorithm give a $(2.836+\epsilon)$-approximation; our result improves upon the $(3+\epsilon)$-approximate local-search…
We design a new algorithm for the Euclidean $k$-means problem that operates in the local model of differential privacy. Unlike in the non-private literature, differentially private algorithms for the $k$-means objective incur both additive…