The NP-hard number-partitioning problem is to separate a multiset S of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized. The classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan, the elapsed time to complete the schedule, is minimized. The two-way number-partitioning decision problem is one of the original 21 problems that Richard Karp proved NP-complete. It is also one of Garey and Johnson’s six fundamental NP-complete problems and the only one based on numbers. This article explores algorithms for solving multi-way number-partitioning problems optimally. We explore previous algorithms as well as our own algorithms, which fall into three categories: sequential number partitioning (SNP), a branch-and-bound algorithm; binary-search improved bin completion (BSIBC), a bin-packing algorithm; and cached iterative weakening (CIW), an iterative weakening algorithm. We show experimentally that, for large random numbers, SNP and CIW are state-of-the-art algorithms depending on the values of n and k . Both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime.

The NP-hard number-partitioning problem is to separate a multiset S of n positive integers into k subsets, such that the largest sum of the integers assigned to any subset is minimized. The classic application is scheduling a set of n jobs with different run times onto k identical machines such that the makespan, the time to complete the schedule, is minimized. We present a new algorithm, cached iterative weakening (CIW), for solving this problem optimally. It incorporates three ideas distinct from the previous state of the art: it explores the search space using iterative weakening instead of branch and bound; generates feasible subsets once and caches them instead of at each node of the search tree; and explores subsets in cardinality order instead of an arbitrary order. The previous state of the art is represented by three different algorithms depending on the values of n and k. We provide one algorithm which outperforms all previous algorithms for k >= 4. Our run times are up to two orders of magnitude faster.

We present an algorithm for compressing pattern databases (PDBs) and a method for fast random access of these com-pressed PDBs. We demonstrate the effectiveness of our technique by compressing two 6-tile sliding-tile PDBs by a factor of 12 and a 7-tile sliding-tile PDB by a factor of 24.