980 research outputs found

    Evaluations of topological Tutte polynomials

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    We find new properties of the topological transition polynomial of embedded graphs, Q(G)Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, R(G)R(G), and the topological Penrose polynomial, P(G)P(G). The general framework provided by Q(G)Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G)P(G), R(G)R(G), and the Tutte polynomial, T(G)T(G), as sums of chromatic polynomials of graphs derived from GG; show that these polynomials count kk-valuations of medial graphs; show that R(G)R(G) counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of R(G)R(G). 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 P(G)P(G) and R(G)R(G).Comment: V2: major revision, several new results, and improved expositio

    Exponential Time Complexity of the Permanent and the Tutte Polynomial

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    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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