Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see [this article][1] by Paul Erdös and Sherman Stein. 

Is there a polynomial-time algorithm that takes two positive integers $m,n$ as input and gives the minimum number of elements of a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$?

  [1]: https://www.renyi.hu/~p_erdos/1963-18.pdf