Approximation algorithms for multi-criteria traveling salesman problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of min{1+γ,2γ²/2γ²−2γ+1}+ε and a randomized approximation algorithm that achieves a ratio of 2γ³+2γ²/3γ2−2γ+1+ε. In particular, we obtain a 2+ε approximation for multi-criteria metric STSP.

Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio 1+γ/1+3γ−4γ²+ε), asymmetric TSP (ATSP) with γ-triangle inequality (ratio 1/2+γ³/1−3γ²+ε ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2).