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

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...

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Vydáno v:Computational optimization and applications Ročník 57; číslo 2; s. 403 - 415
Hlavní autoři: Dickinson, Peter J. C., Gijben, Luuk
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.03.2014
Springer Nature B.V
Témata:
Algorithms
Complexity
Complexity theory
Computation
Convex and Discrete Geometry
Management Science
Mathematical models
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Programming
Proving
Statistics
Studies
Copositive
NP-hard
Stable set
Completely positive
ISSN:0926-6003, 1573-2894
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Shrnutí: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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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-013-9594-z