On Hard Instances of Non-Commutative Permanent

  • Christian Engels
  • B. V. Raghavendra RaoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)


Recent developments on the complexity of the non-commutative determinant and permanent [Chien et al. STOC 2011, Bläser ICALP 2013, Gentry CCC 2014] have settled the complexity of non-commutative determinant with respect to the structure of the underlying algebra. Continuing the research further, we look to obtain more insights on hard instances of non-commutative permanent and determinant.

We show that any Algebraic Branching Program (ABP) computing the Cayley permanent of a collection of disjoint directed two-cycles with distinct variables as edge labels requires exponential size. For graphs where every connected component contains at most six vertices, we show that evaluating the Cayley permanent over any algebra containing \(2\times 2\) matrices is \(\#\mathsf{P}\) complete.

Further, we obtain efficient algorithms for computing the Cayley permanent/determinant on graphs with bounded component size, when vertices within each component are not far apart from each other in the Cayley ordering. This gives a tight upper and lower bound for size of ABPs computing the permanent of disjoint two-cycles. Finally, we exhibit more families of non-commutative polynomial evaluation problems that are complete for \(\#\mathsf{P}\).

Our results demonstrate that apart from the structure of underlying algebras, relative ordering of the variables plays a crucial role in determining the complexity of non-commutative polynomials.


Component Size Arithmetic Circuit Polynomial Size Output Gate Exponential Size 
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.



The authors like to thank V. Arvind and Markus Bläser for helpful discussions and pointing out specific problems to work on. The authors also thank anonymous referees for their comments which helped in improving the presentation. This work was partially done while the first author was visiting IIT Madras sponsored by the Indo-Max-Planck Center for Computer Science.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.IIT MadrasChennaiIndia

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