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. For any ϵ > 0 , we provide a polynomial time approximation algorithm that computes a solution of value at least O P T 1 - ϵ 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 any constant ratio unless P = N P .