Efficient optimization of the Held–Karp lower bound

Given a weighted undirected graph G = (V,E), the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex p 2 V , by computing a minimum cost tree spanning V {p} and adding two minimum cost edges adjacent to p. In general, different selections of vertex p provide different lower bounds. In this paper it is shown that the selection of vertex p can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.