# Helsgaun's $k$-Opt moves

In his 2009 paper General k-opt submoves for the Lin–Kernighan TSP heuristic, Helsgaun defines the local tour improvements on which the LKH heuristics are based as:

several questions arise from that definition of admissible $$k$$-Opt moves:

• Where is the proof that the set of exchanged edges must form a cycle?
• Why aren't adjacent tour edges considered as exchange candidates?

Here is a simple counter example to the neessity of the single-cycle conjecture and to the non-adjacent tour edges assumption plus is it a cheaper $$3$$-Opt move than the ones requiring non-adjacent tour edges:

• One pair of the red tour edges that are exchanged with the solid black edges is adjacent.
• The union of exchanged tour-edges with the replacement edges consists of two vertex-disjoint cycles depicted by the thin ellipses.
• The exchange of the edges resembles a transition between tours.
• I’m voting to close this question because, of your 5 questions, only 2 are mathematical. The others might belong on AcademiaSE. Apr 26 at 15:16