Complexity of the Cover Polynomial

  • Markus Bläser
  • Holger Dell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


The cover polynomial introduced by Chung and Graham is a two-variate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is \(\sharp\)-hard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomial: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial is \(\sharp\)-hard under polynomial-time Turing reductions, while only three points are easy. Our construction uses a gadget which is easier to analyze and more general than the XOR-gadget used by Valiant in his proof that the permanent is \(\sharp\)-complete.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Markus Bläser
    • 1
  • Holger Dell
    • 1
  1. 1.Computational Complexity Group, Saarland UniversityGermany

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