  Solving the Traveling Salesperson Problem (TSP) remains a persistent
challenge, despite its fundamental role in numerous generalized applications in
modern contexts. Heuristic solvers address the demand for finding high-quality
solutions efficiently. Among these solvers, the Lin-Kernighan-Helsgaun (LKH)
heuristic stands out, as it complements the performance of genetic algorithms
across a diverse range of problem instances. However, frequent timeouts on
challenging instances hinder the practical applicability of the solver.
  Within this work, we investigate a previously overlooked factor contributing
to many timeouts: The use of a fixed candidate set based on a tree structure.
Our investigations reveal that candidate sets based on Hamiltonian circuits
contain more optimal edges. We thus propose to integrate this promising
initialization strategy, in the form of POPMUSIC, within an efficient restart
version of LKH. As confirmed by our experimental studies, this refined TSP
heuristic is much more efficient - causing fewer timeouts and improving the
performance (in terms of penalized average runtime) by an order of magnitude -
and thereby challenges the state of the art in TSP solving.
