The problem of constrained clustering has attracted significant attention in the past decades. In this paper, we study the balanced @math -center, @math -median, and @math -means clustering problems where the size of each cluster is constrained by the given lower and upper bounds. The problems are motivated by the applications in processing large-scale data in high dimension. Existing methods often need to compute complicated matchings (or min cost flows) to satisfy the balance constraint, and thus suffer from high complexities especially in high dimension. We develop an effective framework for the three balanced clustering problems to address this issue, and our method is based on a novel spatial partition idea in geometry. For the balanced @math -center clustering, we provide a @math -approximation algorithm that improves the existing approximation factors; for the balanced @math -median and @math -means clusterings, our algorithms yield constant and @math -approximation factors with any @math . More importantly, our algorithms achieve linear or nearly linear running times when @math is a constant, and significantly improve the existing ones. Our results can be easily extended to metric balanced clusterings and the running times are sub-linear in terms of the complexity of @math -point metric.
Motivated by the applications from big data, we assume that both the number of input points @math and dimension @math are large, and propose faster algorithms for @math -BCenter, @math -BMedian, @math -BMeans, and their metric counterparts. In addition, we assume that the number of clusters @math is a constant. Actually, @math is usually a small number in practice (e.g., the data is distributed over less than @math machines). Moreover, the existing research on large-scale clustering problems often assume that either @math or @math is a constant, due to their hardness results mentioned above @cite_22 @cite_33 @cite_34 @cite_2 @cite_55 @cite_18 @cite_30 @cite_36 .
@cite_30 @cite_18 @cite_22 @cite_33 @cite_36 @cite_55 @cite_2 @cite_34
2018165630 2947175317 2901801762 2734881792
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. 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. Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair @math -center problem and a @math -approximation for the fair @math -median problem, where @math is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair @math -center. We extend and improve the known results. Firstly, we give a 5-approximation for the fair @math -center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives @math -center, @math -supplier, @math -median, @math -means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed. 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, @math -median, and @math -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), @math -Median, and @math -Means. In UFL with outliers, we have to open some centres, discard up to @math points of @math and assign every other point to the nearest open centre, minimizing the total assignment cost plus centre opening costs. In @math -Median and @math -Means, we have to open up to @math centres but there are no opening costs. In @math -Means, the cost of assigning @math to @math is @math . We present several results. Our main focus is on cases where @math is a doubling metric or is the shortest path metrics of graphs from a minor-closed family of graphs. For uniform-cost UFL with outliers on such metrics we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the @math -Median and @math -Means problems in the same metrics where, for any constant @math , we can find a solution using @math centres whose cost is at most a @math -factor of the optimum and uses at most @math outliers. We also show that natural local search heuristics that do not violate the number of clusters and outliers for @math -Median (or @math -Means) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for @math -Means with outliers to obtain a @math bicriteria.