A Complete Anytime Algorithm for Number Partitioning

Given a set of numbers, the two-way 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 NPcomplete, and is contained in many scheduling applications. 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 algorithm for large problem instances. By restricting the numbers to twelve signi cant digits, CKK can optimally solve two-way partitioning problems of arbitrary size in practice. For numbers with greater precision, CKK rst returns the Karmarkar-Karp solution, then continues to nd better solutions as time allows. Over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time. CKK is directly applicable to the subset sum problem, by reducing it to number partitioning. 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 e ective for other NP-hard problems as well.