On the cover polynomial of a digraph
Journal of Combinatorial Theory Series B
Approximately Counting Hamilton Paths and Cycles in Dense Graphs
SIAM Journal on Computing
Regular Article: The Cycle-Path Indicator Polynomial of a Digraph
Advances in Applied Mathematics
A Tutte Polynomial for Coloured Graphs
Combinatorics, Probability and Computing
On the Complexity of Computing the Tutte Polynomial of Bicircular Matroids
Combinatorics, Probability and Computing
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
On counting homomorphisms to directed acyclic graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Uniform Algebraic Reducibilities between Parameterized Numeric Graph Invariants
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
An extension of the bivariate chromatic polynomial
European Journal of Combinatorics
Complexity of the Bollobás-Riordan polynomial: exceptional points and uniform reductions
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Exponential time complexity of the permanent and the Tutte polynomial
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
A Complexity Dichotomy for Partition Functions with Mixed Signs
SIAM Journal on Computing
The enumeration of vertex induced subgraphs with respect to the number of components
European Journal of Combinatorics
The complexity of the cover polynomials for planar graphs of bounded degree
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Weighted counting of k-matchings is #w[1]-hard
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Noncommutativity makes determinants hard
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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 #Phard 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 #Phard 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 #P-complete.