Improved and Derandomized Approximations for Two-Criteria Metric Traveling Salesman

We improve and derandomize the best known approximation algorithm for the twocriteria metric traveling salesman problem (2-TSP). More precisely, we construct a deterministic 2-approximation which answers an open question by Manthey. Moreover, we show that 2-TSP is randomized (3/2 + ε, 2)-approximable, and we give the first randomized approximations for the two-criteria traveling salesman path problems 2-TSPP, 2-TSPPs, and 2-TSPPst. We further provide 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). A particular interesting situation emerges for 2-TSP: Because of possible trade-offs between the approximation ratios in the first and in the second component, there could exist randomized approximation algorithms that are incomparable to our algorithms. For these we can narrow down the approximation ratios that could be within reach, i.e., that will not force us to improve the well-studied approximations by Christofides and Hoogeveen. This leads to the question of whether 2-TSP has an (α, β)-approximation where 5/3 ≤ α, β < 2.