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In this paper, we study the hard uniform capacitated k-median problem. We give (5 +) factor approximation for the problem using local search technique, violating cardinality by a factor of 3. Though better results are known for the problem using LP techniques, local search algorithms are well known to be simpler. There is a trade-off viz-a-viz approximation factor and cardinality violation between our result and the result of Korupolu et al. [10] which is the only result known for the problem using local search. They gave (1 + α) approximation factor with (5 + 5/α) factor loss in cardinality. In a sense, our result is an improvement as they violate the cardinality by more than a factor of 6 to achieve 5 factor in approximation. Though in their result, the approximation factor can be made arbitrarily small, cardinality loss is at least 5 and small approximation factor is obtained at a big loss in cardinality. Thus, we improve upon their result with respect to cardinality. Povzetek: Obravnavan je NP problem optimiranja iskanja k median in predlagana izvirna rešitev, ki dosega boljše rezultate v določenih primerjavah.

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Clustering is one of the most fundamental problem in Machine Learning. Researchers in the field often require a lower bound on the size of the clusters to maintain anonymity and upper bound for the ease of analysis. Specifying an optimal cluster size is a problem often faced by scientists. In this paper, we present a framework to obtain constant factor approximations for some prominent clustering objectives, with lower and upper bounds on cluster size. This enables scientists to give an approximate cluster size by specifying the lower and the upper bounds for it. Our results preserve the lower bounds but may violate the upper bound a little.We also reduce the violation in upper bounds for a special case when the gap between the lower and upper bounds is not too small.