The Maximum Labeled Path Problem
Résumé
In this paper, we study the approximability of the Maximum Labeled Path
problem: given a vertex-labeled directed acyclic graph $D,$ find a path
in $D$ that collects a maximum number of distinct labels. Our main results
are a $\sqrt{OPT}$-approximation algorithm for this problem and a
self-reduction showing that any constant ratio approximation algorithm
for this problem can be converted into a PTAS. This last result, combined
with the {\sc APX}-hardness of the problem, shows that the problem cannot be
approximated within a constant ratio unless $P=NP$.