Partitioning Problems with Splittings and Interval Targets

The n-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s, t, u. For each variant, we give a complete picture of its running time. For n = 2, the running time is easy to identify. Our main results consider any fixed integer n ≥ 3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s ≥ n − 2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t ≥ n − 1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u ≥ (n − 2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than n − 2.