Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for k-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9 + ǫ, a ratio that is known to be tight with respect to such methods.We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) to satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of k-means where the underlying metric is not required to be Euclidean and for k-median in Euclidean metrics.

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Mobile ad hoc networks (MANETs) have become very interesting during last years, but the security is the most important problem they suffer from. Asymmetric cryptography is a very useful solution to provide a secure environment in multihop networks where intermediate nodes are able to read, drop or change messages before resending them. However, storing all keys in every node by this approach is inefficient, if practically possible, in large-scale MANETs due to some limitations such as memory or process capability. In this paper, we propose a new probabilistic key management algorithm for large-scale MANETs. To the best of our knowledge, this is the first method which probabilistically uses asymmetric cryptography to manage the keys in MANETs. In this algorithm, we store only a few keys in each node instead of all. We analytically prove that the network will remain connected with a high probability more than 99.99%. Furthermore, we analytically calculate the average path length in the network and show that this parameter will not have a significant increment using our algorithm. All analytical results are also validated by simulation to make them dependable.

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