The fraction $\ \frac{a}{b}\ $ with coprime integers $\ a,b\ $ with $\ 1<a<b\ $ is supposed to be represented as a sum of pairwise distinct egyptian fractions, for example $$\frac{12}{13}=\frac{1}{2}+\frac{1}{3}+\frac{1}{13}+\frac{1}{78}$$ This is a well-known problem. A solution should be optimal, if the largest denominator is as small as possible (for the given fraction). The other denominators need not be optimal, just the largest.
Wikipedia states that there is no known efficient algorithm to determine the optimal solution in this sense. Therefore my question :
Suppose, a representation is given. Can efficiently be checked whether it is optimal in the above sense ?
Heuristics to improve a given representation in the above sense are also welcome.