On the computational complexity of membership problems for the completely positive cone and its dual

Peter James Clair Dickinson, Luuk Gijben

Research output: Contribution to journalArticleAcademicpeer-review

69 Citations (Scopus)

Abstract

Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi (Math. Program. 39(2):117–129, 1987) that the strong membership problem for the copositive cone, that is deciding whether or not a given matrix is in the copositive cone, is a co-NP-complete problem. From this it has long been assumed that this implies that the question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an NP-hard problem. However, the technical details for this have not previously been looked at to confirm that this is true. In this paper it is proven that the strong membership problem for the completely positive cone is indeed NP-hard. Furthermore, it is shown that even the weak membership problems for both of these cones are NP-hard. We also present an alternative proof of the NP-hardness of the strong membership problem for the copositive cone.
Original languageUndefined
Pages (from-to)403-415
Number of pages13
JournalComputational optimization and applications
Volume57
Issue number2
DOIs
Publication statusPublished - Mar 2014

Keywords

  • EWI-25333
  • Completely positive
  • NP-hard
  • IR-93166
  • Stable set
  • METIS-309674
  • Copositive

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