Multi-Way Number Partitioning: an Information-Theoretic View

The number partitioning problem is the problem of partitioning a given list of numbers into multiple subsets so that the sum of the numbers in each subset are as nearly equal as possible. We introduce two closely related notions of the "most informative" and "most compressible" partitions. Most informative partitions satisfy a principle of optimality property. We also give an exact algorithm (based on Huffman coding) with a running time of O(nlog(n)) in input size n to find the most compressible partition

Search Strategies for Optimal Multi-Way Number Partitioning

The number partitioning problem seeks to divide a set of n numbers across k distinct subsets so as to minimize the sum of the largest partition. In this work, we develop a new optimal algorithm for multi-way number partitioning. A critical observation motivating our methodology is that a globally optimal k-way partition may be recursively constructed by obtaining suboptimal solutions to subproblems of size k − 1. We introduce a new principle of optimality that provides necessary and sufficient conditions for this construction, and use it to strengthen the relationship between sequential decompositions by enforcing upper and lower bounds on intermediate solutions. We also demonstrate how to further prune unpromising partial assignments by detecting and eliminating dominated solutions. Our approach outperforms the previous state-of-the-art by up to four orders of magnitude, reducing average runtime on the largest benchmarks from several hours to less than a second