On Approximating Multi-Criteria TSP

TL;DR

This paper introduces approximation algorithms for various multi-criteria TSP variants, achieving near-optimal ratios for symmetric, asymmetric, and minimum versions, applicable for any fixed number of objectives.

Contribution

It provides the first approximation algorithms for almost all multi-criteria TSP variants, including randomized and deterministic approaches with specific ratios.

Findings

Approximation ratio of 2/3 - eps for symmetric Max-STSP

Approximation ratio of 1/2 - eps for asymmetric Max-ATSP

Deterministic 7/27 ratio for bi-criteria Max-STSP

Abstract

We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP). First, we devise randomized approximation algorithms for multi-criteria maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP, where the edge weights have to be symmetric, we devise an algorithm with an approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps. Our algorithms work for any fixed number k of objectives. Furthermore, we present a deterministic algorithm for bi-criteria Max-STSP that achieves an approximation ratio of 7/27. Finally, we present a randomized approximation algorithm for the asymmetric multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm achieves a ratio of log n + eps.