In this paper we investigate the problem of partitioning an input string T in such a way that compressing individually its parts via a base-compressor C gets a compressed output that is shorter than applying C over the entire T at once. This problem was introduced in the context of table compression, and then further elaborated and extended to strings and trees. Unfortunately, the literature offers poor solutions: namely, we know either a cubic-time algorithm for computing the optimal partition based on dynamic programming, or few heuristics that do not guarantee any bounds on the efficacy of their computed partition, or algorithms that are efficient but work in some specific scenarios (such as the Burrows-Wheeler Transform) and achieve compression performance that might be worse than the optimal-partitioning by a $Ω(\sqrt{\log n})$ factor. Therefore, computing efficiently the optimal solution is still open. In this paper we provide the first algorithm which is guaranteed to compute in $O(n \log_{1+\eps}n)$ time a partition of T whose compressed output is guaranteed to be no more than $(1+ε)$-worse the optimal one, where $ε$ may be any positive constant.