Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inapproximability of the Tutte polynomial   The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that
encodes many interesting properties of the graph. We study the complexity of
the following problem, for rationals x and y: take as input a graph G, and
output a value which is a good approximation to T(G;x,y). Jaeger, Vertigan and
Welsh have completely mapped the complexity of exactly computing the Tutte
polynomial. They have shown that this is #P-hard, except along the hyperbola
(x-1)(y-1)=1 and at four special points. We are interested in determining for
which points (x,y) there is a "fully polynomial randomised approximation
scheme" (FPRAS) for T(G;x,y). Under the assumption RP is not equal to NP, we
prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes
x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two
exceptions to this result are the half-line x<-1, y=1 (which is still open) and
the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show
to be equivalent in difficulty to approximately counting perfect matchings. We
give further intractability results for (x,y) in the vicinity of the origin. A
corollary of our results is that, under the assumption RP is not equal to NP,
there is no FPRAS at the point (x,y)=(0,1--lambda) when \lambda>2 is a positive
integer. Thus there is no FPRAS for counting nowhere-zero \lambda flows for
\lambda>2. This is an interesting consequence of our work since the
corresponding decision problem is in P for example for \lambda=6.

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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys