From Symmetry to Asymmetry: Generalizing TSP Approximations by Parametrization

TL;DR

This paper extends classical TSP approximation algorithms to handle asymmetric cases using parameterized methods, providing new approximation guarantees and practical benefits for moderately asymmetric instances.

Contribution

It introduces parameterized generalizations of the tree doubling and Christofides algorithms for ATSP, with new approximation ratios and kernelization techniques.

Findings

Parameterized algorithms achieve constant factor approximations for ATSP.

Experimental results show tree doubling often outperforms Christofides in practice.

Algorithms can be combined with polynomial-time approximations for improved performance.

Abstract

We generalize the tree doubling and Christofides algorithm, the two most common approximations for TSP, to parameterized approximations for ATSP. The parameters we consider for the respective parameterizations are upper bounded by the number of asymmetric distances in the given instance, which yields algorithms to efficiently compute constant factor approximations also for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, where the parameter is the size of a vertex cover for the subgraph induced by the asymmetric edges. Our generalization of the tree doubling algorithm gives a parameterized 3-approximation, where the parameter is the number of asymmetric edges in a given minimum spanning arborescence. Both algorithms are also stated in the form of additive lossy kernelizations, which allows to combine them with known polynomial time approximations for ATSP. Further, we combine them with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. We complement our results by experimental evaluations, which show that generalized tree-doubling frequently outperforms generalized Christofides with respect to parameter size.