We propose a generalized definition for the multi-objective traveling salesman problem which uses multigraphs and which allows multiple visits of cities. The definition has two benefits: it captures typical real-world scenarios and it contains the conventional definition (componentwise metric cost function) as a special case.

We provide approximation algorithms for this general version of the two-objective traveling salesman problem (2-TSP). At the same time, with these algorithms we improve the best known approximations for the conventional case. For 2-TSP we obtain a deterministic 2-approximation, a randomized (32+2) -approximation, and a randomized (322+) -approximation. Moreover, we construct similar algorithms for two-objective traveling salesman path problems.

Further we present arguments that indicate the hardness of improving our randomized approximation algorithms in the sense that such improvements force us to improve the best known approximations for TSP, TSPPs, and TSPPst (Christofides 1976, Hoogeveen 1991). In this way, we can narrow down the approximation ratios for 2-TSP that could be within reach, i.e., that will not immediately improve well-studied approximations. This leads to the question of whether 2-TSP has an () -approximation where 532 .