Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan[2010] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target @math -clustering, the projection of a point @math onto the line joining its cluster center @math and some other center @math , is a large additive factor closer to @math than to @math . This additive factor can be roughly described as @math times the spectral norm of the matrix representing the differences between the given (known) dataset and the means of the (unknown) target clustering. Clearly, the proximity condition implies center separation -- the distance between any two centers must be as large as the above mentioned bound. In this paper we improve upon the work of Kumar and Kannan along several axes. First, we weaken the center separation bound by a factor of @math , and secondly we weaken the proximity condition by a factor of @math . Using these weaker bounds we still achieve the same guarantees when all points satisfy the proximity condition. We also achieve better guarantees when only @math -fraction of the points satisfy the weaker proximity condition. The bulk of our analysis relies only on center separation under which one can produce a clustering which (i) has low error, (ii) has low @math -means cost, and (iii) has centers very close to the target centers. Our improved separation condition allows us to match the results of the Planted Partition Model of McSherry[2001], improve upon the results of [2006], and improve separation results for mixture of Gaussian models in a particular setting.
There has been an extensive line of work on approximation algorithms for the @math -means problem ( @cite_5 @cite_23 @cite_2 @cite_33 @cite_38 @cite_24 ). The current best guarantee is a @math -approximation algorithm of @cite_24 (with a much simpler analysis in @cite_27 ) if polynomial dependence on @math and the dimension @math is desired. For constant @math , @cite_31 give a PTAS for the @math -means problem. Another popular algorithm for @math -means is the Lloyd's heuristics ( @cite_18 ). This heuristics, combined with a careful seeding of centers, has been shown to have good performance if the data is well separated (see @cite_10 ), or to provide @math -approximation in general @cite_8 . The separation-based results of @cite_10 were improved by @cite_9 .
@cite_38 @cite_18 @cite_33 @cite_8 @cite_9 @cite_24 @cite_27 @cite_23 @cite_2 @cite_5 @cite_31 @cite_10
2567804199 2018165630 2199495299 2947175317
An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. We consider @math -median clustering in finite metric spaces and @math -means clustering in Euclidean spaces, in the setting where @math is part of the input (not a constant). For the @math -means problem, show that if the optimal @math -means clustering of the input is more expensive than the optimal @math -means clustering by a factor of @math , then one can achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math by using a variant of Lloyd's algorithm. In this work we substantially improve this approximation guarantee. We show that given only the condition that the @math -means optimal is more expensive than the @math -means optimal by a factor @math for some constant @math , we can obtain a PTAS. In particular, under this assumption, for any @math we achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math , and exponential in @math and @math . We thus decouple the strength of the assumption from the quality of the approximation ratio. We also give a PTAS for the @math -median problem in finite metrics under the analogous assumption as well. For @math -means, we in addition give a randomized algorithm with improved running time of @math . Our technique also obtains a PTAS under the assumption of that all @math approximations are @math -close to a desired target clustering, in the case that all target clusters have size greater than @math and @math is constant. Note that the motivation of is that for many clustering problems, the objective function is only a proxy for the true goal of getting close to the target. From this perspective, our improvement is that for @math -means in Euclidean spaces we reduce the distance of the clustering found to the target from @math to @math when all target clusters are large, and for @math -median we improve the largeness'' condition needed in the work of to get exactly @math -close from @math to @math . Our results are based on a new notion of clustering stability. In k-means clustering we are given a set of n data points in d-dimensional space Rd and an integer k, and the problem is to determine a set of k points in &#211C;d, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the extremely high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9+e)-approximation algorithm. We show that the approximation factor is almost tight, by giving an example for which the algorithm achieves an approximation factor of (9-e). To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice. Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. We complement several recent papers in this line of research by introducing a general method to reduce bias in the data through random projections in a fair'' subspace. We apply this method to densest subgraph and @math -means. For densest subgraph, our approach based on fair projections allows to recover both theoretically and empirically an almost optimal, fair, dense subgraph hidden in the input data. We also show that, under the small set expansion hypothesis, approximating this problem beyond a factor of @math is NP-hard and we show a polynomial time algorithm with a matching approximation bound. We further apply our method to @math -means. In a previous paper, [NIPS 2017] showed that problems such as @math -means can be approximated up to a constant factor while ensuring that none of two protected class (e.g., gender, ethnicity) is disparately impacted. We show that fair projections generalize the concept of fairlet introduced by to any number of protected attributes and improve empirically the quality of the resulting clustering. We also present the first constant-factor approximation for an arbitrary number of protected attributes thus settling an open problem recently addressed in several works.