On Labeled Traveling Salesman Problems

We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both versions of the problem. For the maximization version we give a 1/2-approximation algorithm and show that it is APX-hard. For the minimization version, we show that it is not approximable within n(1-is an element of) for every is an element of > 0. When every color appears in the graph at most r times and r is an increasing function of n the problem is not O(r(1-is an element of))-approximable. For fixed constant r we analyze a polynomial-time (r-+H-r)/2-approximation algorithm (H-r is the r-th harmonic number), and prove APX-hardness. Analysis of the studied algorithms is shown to be tight.