Abstract

The Held-Karp (HK) lower bound is the solution to the linear programming relaxation of the standard integer programming formulation of the traveling salesman problem (TSP). For numbers of cities N up to 30,000 or more it can be computed exactly using the Simplex method and appropriate separation algorithms, and for N up to a million good approximations can be obtained via iterative Lagrangean relaxation techniques first suggested by Held and Karp. In this paper, we consider three applications of our ability to compute/approximate this bound. First, we provide empirical evidence in support of using the HK bound as a stand-in for the optimal tour length when evaluating the quality of near-optimal tours. We show that for a wide variety of randomly generated instance types the optimal tour length averages less than 0.8% over the HK bound, and even for the real-world instances in TSPLIB the gap is almost always less than 2%. Moreover, our data indicates that the HK bound can provide substa...

Citations