Multi-Way Number Partitioning

The number partitioning problem is to divide a given set of integers into a collection of subsets, so that the sum of the numbers in each subset are as nearly equal as possible. While a very efficient algorithm exists for optimal two-way partitioning, it is not nearly as effective for multi-way partitioning. We develop two new linear-space algorithms for multi-way partitioning, and demonstrate their performance on three, four, and five-way partitioning. In each case, our algorithms outperform the previous state of the art by orders of magnitude, in one case by over six orders of magnitude. Empirical analysis of the running times of our algorithms strongly suggest that their asymptotic growth is less than that of previous algorithms. The key insight behind both our new algorithms is that if an optimal k-way partition includes a particular subset, then optimally partitioning the numbers not in that set k−1ways results in an optimal k-way partition.