On the complexity of approximating tsp with neighborhoods and related problems

Shmuel Safra, Oded Schwartz

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that various geometric covering problems related to the Traveling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the Group-Traveling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the Group-Steiner-Tree in the Euclidean plane and the Minimum Watchman Tour and Minimum Watchman Path in 3-D. Some inapproximability factors are also shown for special cases of the above problems, where the size of the sets is bounded. Group-TSP and Group-Steiner-Tree where each neighborhood is connected are also considered. It is shown that approximating these variants to within any constant factor smaller than 2 is NP-hard. For the Group-Traveling Salesman and Group-Steiner-Tree Problems in dimension d, we show an inapproximability factor of O(log (d-1)/d n) under a plausible conjecture regarding the hardness of Hyper-Graph Vertex-Cover.

Original languageEnglish
Pages (from-to)281-307
Number of pages27
JournalComputational Complexity
Volume14
Issue number4
DOIs
StatePublished - Mar 2006

Keywords

  • Approximation
  • Hardness of approximation
  • Inapproximability
  • NP-optimization problems
  • TSP
  • TSP with neighborhoods

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