Related papers: The Hardness of Approximation of Euclidean k-means
Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to compute a set $S \subset V$ of cardinality $\left\lvert S\right\rvert \leq k$ such that the weight of edges inside $S$ is maximized. This is a fundamental NP-hard…
The fair $k$-median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fair violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. As far as we know,…
Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such…
A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$…
Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy…
We are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k (p < k). In addition, we assume that the input contains all possible subsets of size p. Our objective is to find a…
The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^{O(1)} time where k is some parameter of the input. We obtain the following results on…
In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor…
The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the…
It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS~99), that \MC\ admits a PTAS on dense graphs, and more generally, \kCSP\ admits a PTAS on "dense" instances with $\Omega(n^k)$ constraints. In this paper we…
In the Geometric Median problem with outliers, we are given a finite set of points in d-dimensional real space and an integer m, the goal is to locate a new point in space (center) and choose m of the input points to minimize the sum of the…
We study the classic NP-Hard problem of finding the maximum $k$-set coverage in the data stream model: given a set system of $m$ sets that are subsets of a universe $\{1,\ldots,n \}$, find the $k$ sets that cover the most number of distinct…
The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: {quote}…
We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs…
Algorithms often carry out equally many computations for "easy" and "hard" problem instances. In particular, algorithms for finding nearest neighbors typically have the same running time regardless of the particular problem instance. In…
We consider the Minimum-$(k,\rho)$-$\mathrm{Shortcut}$ problem ($\min(k,\rho)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $\rho$ closest…
Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…
Core decomposition is a classic technique for discovering densely connected regions in a graph with large range of applications. Formally, a $k$-core is a maximal subgraph where each vertex has at least $k$ neighbors. A natural extension of…
The input to the Multiway Cut problem is a weighted undirected graph, with nonnegative edge weights, and $k$ designated terminals. The goal is to partition the vertices of the graph into $k$ parts, each containing exactly one of the…
{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…