Search Strategies for Optimal Multi-Way Number Partitioning

Abstract

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.