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.
@cite_30: The Euclidean @math -means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of @math points in Euclidean space @math , and the goal is to choose @math centers in @math so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general @math and a @math -approximation which runs in time @math . At the other extreme, the only known computational complexity result for this problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in @math can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all @math . In this paper we provide the first hardness of approximation for the Euclidean @math -means problem. Concretely, we show that there exists a constant @math such that it is NP-hard to approximate the @math -means objective to within a factor of @math . We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform @math , a known hard vertex cover instance, by taking a graph product with a suitably chosen graph @math , and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
@cite_18: We show that k-means clustering is an NP-hard optimization problem, even if k is fixed to 2.
@cite_22: The problem of clustering a set of points so as to minimize the maximum intercluster distance is studied. An O(kn) approximation algorithm, where n is the number of points and k is the number of clusters, that guarantees solutions with an objective function value within two times the optimal solution value is presented. This approximation algorithm succeeds as long as the set of points satisfies the triangular inequality. We also show that our approximation algorithm is best possible, with respect to the approximation bound, if PZ NP.
@cite_33: In this paper we present a 2-approximation algorithm for the k-center problem with triangle inequality. This result is “best possible” since for any δ < 2 the existence of δ-approximation algorithm would imply that P = NP. It should be noted that no δ-approximation algorithm, for any constant δ, has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O|E| log |E| time, a solution with value no more than twice the k-center optimal value. A by-product of the analysis is an O|E| algorithm that identifies a dominating set in G2, the square of a graph G, the size of which is no larger than the size of the minimum dominating set in the graph G. The key combinatorial object used is called a strong stable set, and we prove the NP-completeness of the corresponding decision problem.
@cite_36: 
@cite_55: In the k-means problem, we are given a finite set S of points in @?^m, and integer k>=1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta (2007) [7].
@cite_2: 
@cite_34: Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.
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 .