On the Approximation Ratio of the 3-Opt Algorithm for the (1,2)-TSP

TL;DR

This paper analyzes the approximation ratios of 2-Opt and 3-Opt algorithms for the (1,2)-TSP, establishing bounds and introducing an improved 3-Opt++ algorithm with a better ratio.

Contribution

It provides exact approximation ratios for 2-Opt, 3-Opt, and introduces the 3-Opt++ algorithm with an improved ratio for the (1,2)-TSP.

Findings

2-Opt has a lower bound of 3/2 for the (1,2)-TSP.

3-Opt achieves an exact ratio of 11/8.

3-Opt++ improves the ratio to 4/3.

Abstract

The (1,2)-TSP is a special case of the TSP where each edge has cost either 1 or 2. In this paper we give a lower bound of $\frac{3}{2}$ for the approximation ratio of the 2-Opt algorithm for the (1,2)-TSP. Moreover, we show that the 3-Opt algorithm has an exact approximation ratio of $\frac{11}{8}$ for the (1,2)-TSP. Furthermore, we introduce the 3-Opt++-algorithm, an improved version of the 3-Opt algorithm for the (1-2)-TSP with an exact approximation ratio of $\frac{4}{3}$.