The travelling salesman and the PQ-tree

R.E. Burkard, V.G. Deineko, G.J. Woeginger

    Research output: Contribution to journalArticleAcademicpeer-review

    15 Citations (Scopus)
    2 Downloads (Pure)


    Let D = (dij) 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 (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 (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.
    Original languageEnglish
    Pages (from-to)613-623
    JournalMathematics of Operations Research
    Issue number3
    Publication statusPublished - 1998


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