Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

TL;DR

This paper develops new approximation algorithms for multi-criteria traveling salesman problems, providing efficient solutions with provable bounds for complex multi-objective optimization scenarios.

Contribution

It introduces deterministic and randomized approximation algorithms with specific ratios for multi-criteria TSP variants, advancing the computational methods for Pareto curve approximation.

Findings

Deterministic polynomial-time algorithm for multi-criteria g-metric STSP with approximation ratio depending on g.

Randomized algorithms achieving approximation ratios for g-metric STSP and ATSP.

New approximation schemes for multi-criteria cycle cover and graph factor problems.

Abstract

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 design a deterministic polynomial-time algorithm for multi-criteria g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate Pareto curves for all 1/2<=g<=1. In particular, we obtain a (2+eps)-approximation for multi-criteria metric STSP. We also present two randomized approximation algorithms for multi-criteria g-metric STSP that achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1 +3g -4g^2) + eps, respectively. Moreover, we present randomized approximation algorithms for multi-criteria g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do this, we design randomized approximation schemes for multi-criteria cycle cover and graph factor problems.