A complete anytime algorithm for number partitioning

摘要

Given a set of numbers, the two-way number partitioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible. The problem is NP-complete. Based on a polynomial-time heuristic due to Karmarkar and Karp, we present a new algorithm, called Complete Karmarkar-Karp (CKK), that optimally solves the general number-partitioning problem, and significantly outperforms the best previously-known algorithms for large problem instances. For numbers with twelve significant digits or less, CKK can optimally solve two-way partitioning problems of arbitrary size in practice. For numbers with greater precision, CKK first returns the Karmarkar-Karp solution, then continues to find better solutions as time allows. Over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time. Rather than building a single solution one element at a time, or modifying a complete solution, CKK constructs subsolutions, and combines them together in all possible ways. This approach may be effective for other NP-hard problems as well.