A Randomized Algorithm for 2-Partition of a Sequence
In the paper we consider one strongly NP-hard problem of partitioning a finite Euclidean sequence into two clusters minimizing the sum over both clusters of intracluster sum of squared distances from clusters elements to their centers. The cardinalities of clusters are assumed to be given. The center of the first cluster is unknown and is defined as the mean value of all points in the cluster. The center of the second one is the origin. Additionally, the difference between the indexes of two consequent points from the first cluster is bounded from below and above by some constants. A randomized algorithm for the problem is proposed. For an established parameter value, given a relative error \(\varepsilon > 0\) and fixed \(\gamma \in (0, 1)\), this algorithm allows to find a \((1 + \varepsilon )\)-approximate solution of the problem with a probability of at least \(1 - \gamma \) in polynomial time. The conditions are established under which the algorithm is polynomial and asymptotically exact.
KeywordsPartitioning Sequence Euclidean space Minimum sum-of-squared distances NP-hardness Randomized algorithm Asymptotic accuracy
This work was supported by the Russian Foundation for Basic Research, project nos. 15-01-00462, 16-31-00186, and 16-07-00168.