Approximation Algorithms for Multi-criteria Traveling Salesman Problems

Abstract

In multi-criteria optimization, several objective functions are to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as optimal. Instead, the aim is to compute so-called Pareto curves. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We are concerned with approximating Pareto curves of multi-criteria traveling salesman problems (TSP). We provide algorithms for computing approximate Pareto curves for the symmetric TSP with triangle inequality (Δ− STSP), symmetric and asymmetric TSP with strengthened triangle inequality (Δ(γ)−STSP and Δ(γ)− ATSP), and symmetric and asymmetric TSP with weights one and two (STSP(1,2) and ATSP(1,2)). We design a deterministic polynomial-time algorithm that computes (1+γ+ e)-approximate Pareto curves for multi-criteria Δ(γ)−STSP for $\gamma \in [\frac 12, 1]$. We also present two randomized approximation algorithms for multi-criteria Δ(γ)−STSP achieving approximation ratios of $\frac{2\gamma^3 + \gamma^2 + 2 \gamma-1}{2\gamma^2} + \varepsilon$ and $\frac{1+\gamma}{1+3 \gamma -- 4 \gamma^2}$ + e, respectively. Moreover, we design randomized approximation algorithms for multi-criteria Δ(γ)−ATSP (ratio $\frac 12+ \frac{\gamma^3}{1-3\gamma^2}$ + e for $\gamma < 1/\sqrt{3}$), STSP(1,2) (ratio 4/3) and ATSP(1,2) (ratio 3/2). The algorithms for Δ(γ)−ATSP, STSP(1,2), and ATSP(1,2) as well as one algorithm for Δ(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.