Efficient Randomized Algorithm for a Vector Subset Problem

Abstract

We introduce a randomized approximation algorithm for NP-hard problem of finding a subset of m vectors chosen from n given vectors in multidimensional Euclidean space $\mathbb {R}^k$ such that the norm of the corresponding sum-vector is maximum. We derive the relation between algorithm’s time complexity, relative error and failure probability parameters. We show that the algorithm implements Polynomial-time Randomized Approximation Scheme (PRAS) for the general case of the problem. Choosing particular parameters of the algorithm one can obtain asymptotically exact algorithm with significantly lower time complexity compared to known exact algorithm. Another set of parameters provides polynomial-time 1 / 2-approximation algorithm for the problem. We also show that the algorithm is applicable for the related (minimization) clustering problem allowing to obtain better performance guarantees than existing algorithms.