Orbits, schemes and dynamic programming procedures for the TSP 4-OPT neighborhood

TL;DR

This paper analyzes the structure of 4-OPT moves in the TSP, grouping them into orbits, and presents two dynamic programming algorithms with different complexities for finding optimal moves.

Contribution

It introduces a novel grouping of 4-OPT moves into orbits and provides efficient algorithms for selecting the best move in the TSP.

Findings

Grouped 25 4-OPT moves into 7 orbits of equivalent moves

Developed a $ heta(n^3)$ algorithm for 4-OPT move selection

Developed a $ heta(n^2)$ algorithm for 4-OPT move selection

Abstract

We discuss the way to group all 25 possible 4-OPT moves into 7 orbits of equivalent moves. We then describe two implementations, one for a $\Theta(n^3)$ algorithm by de Berg's et al. and one of a $\Theta(n^2)$ algorithm by Glover, for finding the best 4-OPT move via dynamic programming.