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](https://www.renyi.hu/~p_erdos/1963-18.pdf) 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 $\frac{m}{n} = \sum\_{u\in S} u$?



Positive rational numbers as sum of unit fractions
This is the Egyptian fractions problem, and there are a [number of algorithms](http://kevingong.com/Math/EgyptianFractions.pdf). [Wikipedia](http://en.wikipedia.org/wiki/Egyptian_fraction) claims the computational complexity is unknown.


