Core Concepts
The authors study three variants of the classic n-way number partitioning problem that relax the constraints of the original problem. The first two variants allow a bounded number of split items or splittings, while the third variant requires the largest bin sum to be within a pre-specified interval. The authors provide a complete picture of the computational complexity of these variants, showing that they can be solved efficiently in polynomial time in certain parameter regimes, while remaining NP-complete in others.
Abstract
The authors consider three variants of the classic n-way number partitioning problem:
- Dec-SplitItemn, s: Decide if there exists a partition of the items X among n bins with at most s split items, such that the maximum bin sum is at most S.
- Dec-Splittingn, t: Decide if there exists a partition of the items X among n bins with at most t splittings, such that the maximum bin sum is at most S.
- Dec-Intern, u: Decide if there exists a partition of the items X among n bins such that the maximum bin sum is between S and S + u * M, where S is the average bin sum and M is the size of the largest item.
For the Dec-SplitItem and Dec-Splitting variants, the authors show that:
- When the number of split items or splittings is less than n-2 or n-1 respectively, the problems are NP-complete.
- When the number of split items or splittings is at least n-2 or n-1 respectively, the problems can be solved in polynomial time.
For the Dec-Inter variant, the authors show that:
- When u < n-2, the problem is NP-complete.
- When u ≥ n-2, the problem can be solved in polynomial time.
The authors also provide a fully polynomial-time approximation scheme (FPTAS) for the case where the number of split items is lower than n-2.