The travelling salesman and the PQ-tree

Let D = (d(ij)) be the n x n distance matrix of a set of n cities {1, 2,..., n}, and let T be a PQ-tree with node degree bounded by d that represents a set n(T) of permutations over {1, 2,..., n }. We show how to compute for D in O(2(d)n(3)) time the shortest travelling salesman tour contained in n(T). Our algorithm may be interpreted as a common generalization of the well-known Held and Karp dynamic programming algorithm for the TSP and of the dynamic programming algorithm for finding the shortest pyramidal TSP tour. A consequence of our result is that the shortcutting phase of the "twice around the tree" heuristic for the Euclidean TSP can be optimally implemented in polynomial time. This contradicts a statement of Papadimitriou and Vazirani as published in 1984.