The Maximum Labeled Path Problem

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 APX-hardness of the problem, shows that the problem cannot be approximated within a constant ratio unless \(P=NP\).