Multi-Criteria Traveling Salesman Problems

In multi-criteria optimization problems, several objective functions have to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as the optimal solution. Instead, the aim is to compute a so-called Pareto curve of solutions. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We are concerned with multi-criteria versions of several variants of the traveling salesman problem (TSP). We provide polynomial time deterministic and randomized algorithms for computing approximate Pareto curves for the symmetric TSP with triangle inequality (denoted by ∆ -STSP), symmetric and asymmetric TSP with strengthened triangle inequality (∆(γ) -STSP and ∆(γ) -ATSP), symmetric TSP with weights one and two (STSP(1, 2)), and asymmetric TSP with weights one and two (ATSP(1, 2)). We design a deterministic polynomial-time algorithm that computes (1 + γ+ ε)-approximate Pareto curves for multi-criteria ∆(γ) -STSP for γ ∈ [12 , 1]. We also present two randomized approximation algorithms for multi-criteria ∆(γ) -STSP achieving approximation ratios of 2γ 3+γ2+2γ−1 2γ +ε and 1+γ 1+3γ−4γ2 + ε, respectively. Moreover, we present randomized approximation algorithms for multicriteria ∆(γ) -ATSP (ratio 12 + γ 1−3γ2 + ε for γ < 1/ √ 3), STSP(1, 2) (ratio 4/3) and ATSP(1, 2) (ratio 3/2). The algorithms for ∆(γ) -ATSP, STSP(1, 2), and ATSP(1, 2) as well as one algorithm for ∆(γ) -STSP are based on cycle covers. Therefore, we design randomized approximation schemes for multi-criteria cycle cover problems by showing that multi-criteria graph factor problems admit fully polynomial-time randomized approximation schemes.