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

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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.

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Acknowledgements

We wish to thank Gerhard J. Woeginger, Mirjam Dür and the anonymous referees for their highly useful advice and comments on this paper. Luuk Gijben was supported by the Netherlands Organisation for Scientific Research (NWO) through grant no.639.033.907. This research was carried out whilst Peter J.C. Dickinson was a PhD student at the Johann Bernoulli Institute, University of Groningen.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Peter J. C. Dickinson

  2. Johann Bernoulli Institute, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Luuk Gijben

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  1. Peter J. C. Dickinson
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  2. Luuk Gijben
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Correspondence to Peter J. C. Dickinson.

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Dickinson, P.J.C., Gijben, L. On the computational complexity of membership problems for the completely positive cone and its dual. Comput Optim Appl 57, 403–415 (2014). https://doi.org/10.1007/s10589-013-9594-z

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