  We prove that the absolute extendability constant of a finite metric space
may be determined by computing relative projection constants of certain
Lipschitz-free spaces. As an application, we show that $\mbox{ae}(3)=4/3$ and
$\mbox{ae}(4)\geq (5+4\sqrt{2})/7$. Moreover, we discuss how to compute
relative projection constants by solving linear programming problems.
