Efficient optimization of the Held–Karp lower bound

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