A domination algorithm for {0,1}‐instances of the travelling salesman problem

Kühn, Daniela; Osthus, Deryk; Patel, Viresh

doi:10.1002/rsa.20600

Cited by 3 publications

(2 citation statements)

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“…Travelling Salesman Problem Below Average (TSP BA ) Instance : (n, w, k), where n, k ∈ N and w : E(K n ) → Z Question : Is there a Hamilton cycle H * of K n satisfying w(H * ) ≤ dn−k? Theorem 1.1 is proved by applying a combination of probabilistic, combinatorial, and algorithmic techniques, some of which are inspired by [11].…”

Section: (And Outputs Such a Hamilton Cycle If It Exists) In Timementioning

confidence: 99%

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Parameterized TSP: Beating the Average

Gutin,

Patel

2014

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In the Travelling Salesman Problem (TSP), we are given a complete graph K n together with an integer weighting w on the edges of K n , and we are asked to find a Hamilton cycle of K n of minimum weight. Let h(w) denote the average weight of a Hamilton cycle of K n for the weighting w. Vizing (1973) asked whether there is a polynomialtime algorithm which always finds a Hamilton cycle of weight at most h(w). He answered this question in the affirmative and subsequently Rublineckii (1973) and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalisation of Vizing's result: for each fixed k, we give an algorithm that decides whether, for any input edge weighting w of K n , there is a Hamilton cycle of K n of weight at most h(w) − k (and constructs such a cycle if it exists). For k fixed, the running time of the algorithm is polynomial in n, where the degree of the polynomial does not depend on k (i.e., the generalised Vizing problem is fixed-parameter tractable with respect to the parameter k).

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Section: (And Outputs Such a Hamilton Cycle If It Exists) In Timementioning

confidence: 99%

“…By an algorithmic version of Dirac's Theorem (see e.g. [11] Lemma 5.11), we can find, in time O(n 3 ), a Hamilton cycle H of K 0 w [X ′ ] ∪ {e 1 , . .…”

Section: Then We Can Find a Minimum Weight Hamilton Cyclementioning

confidence: 99%