I would like to have a criterion for an extension of the egyptian-fraction representation (Egyptian fractions have numerator $1$). I allow negative fractions, but the occuring denominators have to be distinct.
For example, $\frac{8}{11}$ , for which I could not find a representation $$\frac{8}{11}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ with positive distinct integers $a,b,c$ (How can I proof that there is none ?), allows the representation $$\frac{8}{11}=\frac{1}{2}+\frac{1}{4}-\frac{1}{44}$$ which is superior compared to $$\frac{8}{11}=\frac{1}{2}+\frac{1}{6}+\frac{1}{22}+\frac{1}{66}$$ even in two ways : The length is smaller AND the absolute value of the largest denominator is smaller.
Even a fraction like $\frac{36}{457}$ allows the representation $$\frac{1}{13}+\frac{1}{540}-\frac{1}{3208140}$$ The greedy algorithm (always choosing the largest possible fraction) would lead to a long representation with very large denominators.
Does someone know a relatively easy criterion whether a given fraction $\frac{a}{b}$ with positive integers $0<a<b$ and $gcd(a,b)=1$ can be represented as a sum of $k$ egyptian fractions (as said, negative denominators are allowed, but no duplicate absolute values of the denominators) ? I am especially interested in the cases $k=2$ and $k=3$.