The travelling salesman and the PQ-tree

Abstract

Let D = (dij) be the n × 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 II(T) of permutations over {1, 2,..., n}. We show how to compute for D in O(2dn3) time the shortest travelling salesman tour contained in II(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. This result has two surprising consequences. The first consequence concerns large sets of permutations, so-called exponential neighborhoods, over which the TSP can be solved efficiently. Up to now, the largest known, neighborhoods had cardinality 2Θ(n), whereas our result yields new neighborhoods of cardinality 2Θ(nloglogn). The second consequence 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.