# Is there an FPTAS for 3-way number partitioning?

The maximization problem of the 3-way number partitioning reads as follows: given $$n$$ positive integers, partition them into 3 subsets such that the smallest sum is as large as possible. It is known to be NP-hard, but does it have an FPTAS?

I found answers for some related problems:

Is there any FPTAS for 3-way number partitioning?

I found an FPTAS for a very similar problem: finding a partition such that the largest sum is as small as possible. This problem is equivalent to the problem of identical-machines scheduling: there are $$m$$ identical machines (in our case $$m=3$$), and $$n$$ jobs with different processing times, and we need to assign jobs to machines such that the latest completion time is as early as possible. In this paper:
there is an algorithm that attains at most $$1+\epsilon$$ of the minimum in time $$O(n\cdot (n^2 / \epsilon)^{m-1})$$, which is an FPTAS for any fixed $$m$$.