Probabilistic analysis of the Held and Karp lower bound for the Euclidean traveling salesman problem

The authors analyze probabilistically the classical Held-Karp lower bound derived from the 1-tree relaxation for the Euclidean traveling salesman problem (ETSP). They prove that, if n points are identically and independently distributed according to a distribution with bounded support and absolutely continuous part dx over the d-cube, the Held-Karp lower bound on these n points is almost surely asymptotic to , where is a constant independent of n. The result suggests a probabilistic explanation of the observation that the lower bound is very close to the length of the optimal tour in practice, since the ETSP is almost surely asymptotic to . The techniques the authors use exploit the polyhedral description of the Held-Karp lower bound and the theory of subadditive Euclidean functionals.