The Consensus Clustering problem has been introduced as an effective way to analyze the results of different microarray experiments. The problem consists of looking for a partition that best summarizes a set of input partitions (each corresponding to a different microarray experiment) under a simple and intuitive cost function. The problem admits polynomial time algorithms on two input partitions, but is APX-hard on three input partitions. We investigate the restriction of Consensus Clustering when the output partition is required to contain at most k sets, giving a polynomial time approximation scheme (PTAS) while proving the NP-hardness of this restriction.
We give an improved analysis of the simple $D^2$-sampling based PTAS for the $k$-means clustering problem given by Jaiswal, Kumar, and Sen (Algorithmica, 2013). The improvement on the running time is from $O\left(nd \cdot 2^{\tilde{O}(k^2/ε)}\right)$ to $O\left(nd \cdot 2^{\tilde{O}(k/ε)}\right)$.
Given $n$ length-$\ell$ strings $S =\{s_1, ..., s_n\}$ over a constant size alphabet $Σ$ together with parameters $d$ and $k$, the objective in the {\em Consensus String with Outliers} problem is to find a subset $S^*$ of $S$ of size $n-k$ and a string $s$ such that $\sum_{s_i \in S^*} d(s_i, s) \leq d$. Here $d(x, y)$ denotes the Hamming distance between the two strings $x$ and $y$. We prove 1. a variant of {\em Consensus String with Outliers} where the number of outliers $k$ is fixed and the objective is to minimize the total distance $\sum_{s_i \in S^*} d(s_i, s)$ admits a simple PTAS. (ii)…
Given a set of points $P \subset \mathbb{R}^d$, the $k$-means clustering problem is to find a set of $k$ {\em centers} $C = \{c_1,...,c_k\}, c_i \in \mathbb{R}^d,$ such that the objective function $\sum_{x \in P} d(x,C)^2$, where $d(x,C)$ denotes the distance between $x$ and the closest center in $C$, is minimized. This is one of the most prominent objective functions that have been studied with respect to clustering. $D^2$-sampling \cite{ArthurV07} is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points $P \subseteq \math…
We generalise the results of Bhattacharya et al. (Journal of Computing Systems, 62(1):93-115, 2018) for the list-$k$-means problem defined as -- for a (unknown) partition $X_1, ..., X_k$ of the dataset $X \subseteq \mathbb{R}^d$, find a list of $k$-center sets (each element in the list is a set of $k$ centers) such that at least one of $k$-center sets $\{c_1, ..., c_k\}$ in the list gives an $(1+\varepsilon)$-approximation with respect to the cost function $\min_{\textrm{permutation } π} \left[ \sum_{i=1}^{k} \sum_{x \in X_i} ||x - c_{π(i)}||^2 \right]$. The list-$k$-means problem is important…
In this work, we study the $k$-median clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. Our main result is the first lossy ($2$-approximate) polynomial kernel for this problem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS.
We study clustering problems such as k-Median, k-Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) [Feldmann et al. SICOMP 2018] or run in FPT time (parameterized by the number of clusters $k$, the highway dimension, and the approximation factor) [Becker et al. ESA~2018, Braverman et al. 2020]. In this paper we show that a polynomial-time approximation scheme (PTAS…
Motivated by applications in redistricting, we consider the uniform capacitated k-median and uniform capacitated k-means problems in bounded doubling metrics. We provide the first QPTAS for both problems and the first PTAS for the uniform capacitated k-median problem for points in R^2 . This is the first improvement over the bicriteria QPTAS for capacitated k-median in low-dimensional Euclidean space of Arora, Raghavan, Rao [STOC 1998] (1 + ε-approximation, 1 + ε-capacity violation) and arguably the first polynomial-time approximation algorithm for a non-trivial metric.
\textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{$k$-means} clustering has got much attention from the researchers. Despite the fact that $k$-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is $9+\eps$. In this paper, we consider t…
We revisit the problem of finding a minimum enclosing ball with differential privacy: Given a set of $n$ points in the Euclidean space $\mathbb{R}^d$ and an integer $t\leq n$, the goal is to find a ball of the smallest radius $r_{opt}$ enclosing at least $t$ input points. The problem is motivated by its various applications to differential privacy, including the sample and aggregate technique, private data exploration, and clustering. Without privacy concerns, minimum enclosing ball has a polynomial time approximation scheme (PTAS), which computes a ball of radius almost $r_{opt}$ (the problem…
Phylogenetic trees (i.e. evolutionary trees, additive trees or X-trees) play a key role in the processes of modeling and representing species evolution. Genome evolution of a given group of species is usually modeled by a species phylogenetic tree that represents the main patterns of vertical descent. However, the evolution of each gene is unique. It can be represented by its own gene tree which can differ substantially from a general species tree representation. Consensus trees and supertrees have been widely used in evolutionary studies to combine phylogenetic information contained in indivi…
The most well known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centres and partition the data points into $k$ clusters around these centres so that the sum of squares of distances of the points to their cluster centre is minimized. Typically these data points lie $\mathbb{R}^d$ for some $d\geq 2$. $k$-means and the first algorithms for it were introduced in the 1950's. Since then, hundreds of papers have studied this problem and many algorithms have been proposed for it. Th…
In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph $G$ with `+' and `-' edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all `+' edges within clusters plus all `-' edges cut across clusters. BCC is known to be NP-hard. We present a novel approximation algorithm for $k$-BCC, a variant of BCC with an upper bound $k$ on the number of clusters. Our algorithm outputs a $k$-clustering that provably achieves a number of agreements within a multiplicative ${(1-δ)}$-factor from the optimal, for any desired accuracy $δ$. It reli…
Ashtiani et al. (NIPS 2016) introduced a semi-supervised framework for clustering (SSAC) where a learner is allowed to make same-cluster queries. More specifically, in their model, there is a query oracle that answers queries of the form given any two vertices, do they belong to the same optimal cluster?. Ashtiani et al. showed the usefulness of such a query framework by giving a polynomial time algorithm for the k-means clustering problem where the input dataset satisfies some separation condition. Ailon et al. extended the above work to the approximation setting by giving an efficient (1+\ep…
We study the problem of organizing a collection of objects - images, videos - into clusters, using crowdsourcing. This problem is notoriously hard for computers to do automatically, and even with crowd workers, is challenging to orchestrate: (a) workers may cluster based on different latent hierarchies or perspectives; (b) workers may cluster at different granularities even when clustering using the same perspective; and (c) workers may only see a small portion of the objects when deciding how to cluster them (and therefore have limited understanding of the "big picture"). We develop cost-effi…
Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, $k$-median, and $k$-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), $k$-Median, and $k$-Means. In UFL with outliers, we have to open some centres, discard up to $z$ points of $\cal X$ and assign every other point to the nearest o…
A Fixed Parameter Sublinear Time Approximation Scheme for Metric Min-Sum Clustering
Rutgers University（罗格斯大学）；University of Michigan（密歇根大学）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Aarhus University（奥胡斯大学）；Texas A&M University（德克萨斯农工大学）
A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimension ($N_{_{D}}>3$). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clustering techniques are used, including spectral clustering, however, new techniques are also introduced based on the path length between partitions that are connected to one another. A Line-Of-Sight algorithm is also developed for clustering. A test bank of 12 data sets with varying properties is used to expose the strengths and weaknesses of each technique. Finally, a ro…
Yahoo! Research（雅虎研究院）；IAS（高等研究院）；Technion Institute of Technology（以色列理工学院）；Princeton University（普林斯顿大学）
This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in Rd, the objective is to produce the hyperplane (passing through the origin) which maximizes the separation margin, that is, the minimal distance between the hyperplane and any input point. To the best of our knowledge, this is the first paper achieving provable results regarding FHP. We provide both lower and upper bounds to this NP-hard problem. First, we give a simple randomized algorithm whose running time is n^O(1/θ^2) where θ is the optim…
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are \textsc{Low GF(2)-Rank Approximation}, \textsc{Low Boolean-Rank Approximation}, and various versions of \textsc{Binary Clu…
The input to the $k$-median for lines problem is a set $L$ of $n$ lines in $\mathbb{R}^d$, and the goal is to compute a set of $k$ centers (points) in $\mathbb{R}^d$ that minimizes the sum of squared distances over every line in $L$ and its nearest center. This is a straightforward generalization of the $k$-median problem where the input is a set of $n$ points instead of lines. We suggest the first PTAS that computes a $(1+ε)$-approximation to this problem in time $O(n \log n)$ for any constant approximation error $ε\in (0, 1)$, and constant integers $k, d \geq 1$. This is by proving that ther…
