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The n-way number partitioning problem, a fundamental challenge incombinatorial optimization, has significant implications for applications suchas fair division and machine scheduling. Despite these problems being NP-hard,many approximation techniques exist. We consider three closely related kinds ofapproximations, and various objectives such as decision, min-max, max-min, andeven a generalized objective, in which the bins are not considered identicalanymore, but rather asymmetric (used to solve fair division to asymmetricagents or uniform machine scheduling problems).  The first two variants optimize the partition such that: in the first variantsome fixed number s of items can be split between two or more bins and in thesecond variant we allow at most a fixed number t of splittings. The thirdvariant is a decision problem: the largest bin sum must be within apre-specified interval, parameterized by a fixed rational number u times thelargest item size.  When the number of bins n is unbounded, we show that every variant isstrongly NP-complete. When the number of bins n is fixed, the running timedepends on the fixed parameters s,t,u. For each variant, we give a completepicture of its running time.  For n=2, the running time is easy to identify. Our main results consider anyfixed n>=3. Using a two-way polynomial-time reduction between the first and thethird variant, we show that n-way number-partitioning with s split items can besolved 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 ift>=n-1, and it is NP-complete otherwise. Finally, we show that the thirdvariant can be solved in polynomial time if u>=(n-2)/n, and it is NP-completeotherwise. Our positive results for the optimization problems consider bothasymmetric min-max and asymmetric max-min versions.

Asymmetric Number Partitioning with Splitting and Interval Targets