(Non) -Approximability for the multi-criteria TSP (1,2)

The approximability of multi-criteria combinatorial problems motivated a lot of articles. However, the non-approximability of this problems has never been investigated up to our knowledge. We propose a way to get some results of this kind which works for several problems and we put it into practice on a multi-criteria version of the traveling salesman problem with distances one and two (TSP(1, 2)). Following the article of Angel et al. [1] who presented an approximation algorithm for the bi-criteria TSP(1, 2), we extend and improve the result to any number k of criteria