For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum\_{s\in S} s=\frac{p}{q}$, see [this article](https://www.renyi.hu/~p_erdos/1963-18.pdf) by Paul Erdös and Sherman Stein *(Sums of distinct unit fractions. Proceedings of the American Mathematical Society, 14(1), 126-131, 1963.)* . Let $m(p,q)$ denote the minimal cardinality of such a subset $S$. Is there a polynomial-time algorithm to determine $m(p,q)$?



Minimum number of unit fractions to sum up a given positive rational
In fact, this is listed as an open problem in the [Wikipedia page on Egyptian Fractions](http://en.wikipedia.org/wiki/Egyptian_fraction), presumably because they do use the output size as a parameter.


