We study local search algorithms for metric instances of facility location problems: the uncapacitated facility location problem (UFL), as well as uncapacitated versions of the @math -median, @math -center and @math -means problems. All these problems admit natural local search heuristics: for example, in the UFL problem the natural moves are to open a new facility, close an existing facility, and to swap a closed facility for an open one; in @math -medians, we are allowed only swap moves. The local-search algorithm for @math -median was analyzed by (SIAM J. Comput. 33(3):544-562, 2004), who used a clever coupling'' argument to show that local optima had cost at most constant times the global optimum. They also used this argument to show that the local search algorithm for UFL was 3-approximation; their techniques have since been applied to other facility location problems. In this paper, we give a proof of the @math -median result which avoids this coupling argument. These arguments can be used in other settings where the arguments have been used. We also show that for the problem of opening @math facilities @math to minimize the objective function @math , the natural swap-based local-search algorithm is a @math -approximation. This implies constant-factor approximations for @math -medians (when @math ), and @math -means (when @math ), and an @math -approximation algorithm for the @math -center problem (which is essentially @math ).
The @math -means problem minimizes @math , and is widely used for clustering in machine learning, especially when the point set is in the Euclidean space. For Euclidean instances, one can obtain @math -approximations in linear time, if one imagines @math and @math to be constants: see @cite_7 and the references therein. The most commonly used algorithm in practice is Lloyd's algorithm, which is a local-search procedure different from ours, and which is a special case of the EM algorithm @cite_27 . While there is no explicit mention of an approximation algorithm with provable guarantees for @math -means (to the best of our knowledge), many of the constant-factor approximations for @math -median can be extended to the @math -means problem as well. The paper of @cite_8 is closely related to ours: it analyzes the same local search algorithm we consider, and uses properties of k-means in Euclidean spaces to obtain a @math -approximation. Our results for hold for general metrics, and can essentially be viewed as extensions of their results.
@cite_27 @cite_7 @cite_8
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It has long been realized that in pulse-code modulation (PCM), with a given ensemble of signals to handle, the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall. It has been shown by Panter and Dite that, in the limit as the number of quanta becomes infinite, the asymptotic fractional density of quanta per unit voltage should vary as the one-third power of the probability density per unit voltage of signal amplitudes. In this paper the corresponding result for any finite number of quanta is derived; that is, necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy. The optimization criterion used is that the average quantization noise power be a minimum. It is shown that the result obtained here goes over into the Panter and Dite result as the number of quanta become large. The optimum quautization schemes for 2^ b quanta, b=1,2, , 7 , are given numerically for Gaussian and for Laplacian distribution of signal amplitudes. We present the first linear time (1 + spl epsiv )-approximation algorithm for the k-means problem for fixed k and spl epsiv . Our algorithm runs in O(nd) time, which is linear in the size of the input. Another feature of our algorithm is its simplicity - the only technique involved is random sampling. 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.