Correlation clustering is perhaps the most natural formulation of clustering. Given @math objects and a pairwise similarity measure, the goal is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its theoretical appeal, the practical relevance of correlation clustering still remains largely unexplored, mainly due to the fact that correlation clustering requires the @math pairwise similarities as input. In this paper we initiate the investigation into algorithms for correlation clustering. In we are given the identifier of a single object and we want to return the cluster to which it belongs in some globally consistent near-optimal clustering, using a small number of similarity queries. Local algorithms for correlation clustering open the door to algorithms, which are particularly useful when the similarity between items is costly to compute, as it is often the case in many practical application domains. They also imply @math distributed and streaming clustering algorithms, @math constant-time estimators and testers for cluster edit distance, and @math property-preserving parallel reconstruction algorithms for clusterability. Specifically, we devise a local clustering algorithm attaining a @math -approximation in time @math independently of the dataset size. An explicit approximate clustering for all objects can be produced in time @math (which is provably optimal). We also provide a fully additive @math -approximation with local query complexity @math and time complexity @math . The latter yields the fastest polynomial-time approximation scheme for correlation clustering known to date.
Approximation algorithms for MIN-2-CSP problems @cite_38 also give estimators for cluster edit distance. However, they provide no way of computing each variable assignment in constant time. Moreover, they use time @math to calculate all assignments, and hence do not lend themselves to sublinear-time clustering algorithms.
@cite_38
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One of the fundamental clustering problems is to assign @math points into @math clusters based on minimal sum-of-squared distances (MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP) problem. We show that our 0-1 SDP model provides a unified framework for several clustering approaches such as normalized k-cut and spectral clustering. Moreover, the 0-1 SDP model allows us to solve the underlying problem approximately via the linear programming and SDP relaxations. Second, we consider the issue of how to extract a feasible solution of the original 0-1 SDP model from the optimal solution of the relaxed SDP problem. By using principal component analysis, we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem. In our rounding procedure, we need to solve a K-means clustering problem in @math , which can be done in @math time. In case of biclustering, the running time of our rounding procedure can be reduced to @math . We show that our algorithm provides a 2-approximate solution to the original problem. Promising numerical results for biclustering based on our new method are reported. In a clustering problem, the aim is to partition a given set of n points in d -dimensional space into k groups, called clusters, so that points within each cluster are near each other. Two objective functions frequently used to measure the performance of a clustering algorithm are, for any L 4 metric, (a) the maximum distance between pairs of points in the same cluster, and (b) the maximum distance between points in each cluster and a chosen cluster center; we refer to either measure as the cluster size. We show that one cannot approximate the optimal cluster size for a fixed number of clusters within a factor close to 2 in polynomial time, for two or more dimensions, unless P=NP. We also present an algorithm that achieves this factor of 2 in time O ( n log k ), and show that this running time is optimal in the algebraic decision tree model. For a fixed cluster size, on the other hand, we give a polynomial time approximation scheme that estimates the optimal number of clusters under the second measure of cluster size within factors arbitrarily close to 1. Our approach is extended to provide approximation algorithms for the restricted centers, suppliers, and weighted suppliers problems that run in optimal O ( n log k ) time and achieve optimal or nearly optimal approximation bounds. By the breakthrough work of Hastad [J ACM 48(4) (2001), 798–859], several constraint satisfaction problems are now known to have the following approximation resistance property: Satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin, and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm [J ACM 42(6) (1995), 1115–1145], we know that every Boolean 2-CSP has a nontrivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 − 1 d2). In an attempt to make progress towards this question, in this paper we prove, first, that a slight restriction of this problem, namely, a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, second, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a nontrivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004 We present a novel analysis of a random sampling approach for four clustering problems in metric spaces: k-median, k-means, min-sum k-clustering, and balanced k-median. For all these problems, we consider the following simple sampling scheme: select a small sample set of input points uniformly at random and then run some approximation algorithm on this sample set to compute an approximation of the best possible clustering of this set. Our main technical contribution is a significantly strengthened analysis of the approximation guarantee by this scheme for the clustering problems.The main motivation behind our analyses was to design sublinear-time algorithms for clustering problems. Our second contribution is the development of new approximation algorithms for the aforementioned clustering problems. Using our random sampling approach, we obtain for these problems the first time approximation algorithms that have running time independent of the input size, and depending on k and the diameter of the metric space only. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006A preliminary extended abstract of this work appeared in Proceedings of the 31st Annual International Colloquium on Automata, Languages and Programming (ICALP), pp. 396-407, 2004.