980 research outputs found
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . We conclude with a reduction formula for the
transition polynomial of the tensor product of two embedded graphs, showing
that it leads to additional relations among these polynomials and to further
combinatorial interpretations of and .Comment: V2: major revision, several new results, and improved expositio
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
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