Abstract
The fermionant \({\rm Ferm}^k_n(\bar x) = \sum_{\sigma \in S_n} (-k)^{c(\pi)}\prod_{i=1}^n x_{i,j}\) can be seen as a generalization of both the permanent (for k = − 1) and the determinant (for k = 1). We demonstrate that it is \(\textsc{VNP}\)-complete for any rational k ≠ 1. Furthermore it is #P-complete for the same values of k. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n ε boxes at the right of the first column is \(\textsc{VNP}\)-complete.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Brylinski, J.-L., Brylinski, R.: Complexity and Completeness of Immanants. CoRR, cs.CC/0301024 (2003)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Springer (2000)
Chandrasekharan, S., Wiese, U.-J.: Partition Functions of Strongly Correlated Electron Systems as ”Fermionants”. ArXiv e-prints (August 2011)
de Rugy-Altherre, N.: A dichotomy theorem for homomorphism polynomials, pp. 308–322 (2012)
Greene, C.: A rational-function identity related to the Murnaghan-Nakayama formula for the characters of Sn. J. Algebraic Comb. 1(3), 235–255 (1992)
Littlewood, D.E.: The theory of group characters and matrix representations of groups. The Clarendon Press (1940)
Mertens, S., Moore, C.: The complexity of the fermionant, and immanants of constant width. ArXiv e-prints (October 2011)
Poizat, B.: À la recherche de la définition de la complexité d’espace pour le calcul des polynômes à la manière de valiant. Journal of Symbolic Logic 73 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Rugy-Altherre, N. (2013). Determinant versus Permanent: Salvation via Generalization?. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-39053-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39052-4
Online ISBN: 978-3-642-39053-1
eBook Packages: Computer ScienceComputer Science (R0)