Improved large-step Markov chain variants for the symmetric TSP

The large-step Markov chain (LSMC) approach is the most effective known heuristic for large symmetric TSP instances. In this paper, we examine relationships among (i) the underlying local optimization engine within the LSMC approach, (ii) the ‘kick move’ perturbation that is applied between successive local search descents, and (iii) the resulting LSMC solution quality. We find that the traditional ‘double-bridge’ kick move is not necessarily optimum: stronger local optimization engines are best matched with stronger kick moves. We also propose use of an adaptive temperature schedule to allow escape from deep basins of attraction; the resulting hierarchical LSMC variant outperforms traditional LSMC implementations that use uniformly zero temperatures. Finally, a population-based LSMC variant is studied, wherein multiple solution paths can interact to achieve improved solution quality.