Input: | Polytope P in H-description |
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Output: | The volume of P |

Status (general): | #P-hard, FPRAS |
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Status (fixed dim.): | Polynomial time |

Dyer and Frieze [15] showed that the general problem is
#P-hard (and #P-easy as well). Dyer, Frieze, and
Kannan [16] found a Fully Polynomial Randomized
Approximation Scheme (FPRAS) for the problem, i.e., a
family (A of randomized
algorithms, where, for each _{e})_{e>0}e> 0, A
computes a number _{e}V with the property that the
probability of _{e}(1-e) vol(P) £ V is at least _{e} £ (1+e) vol(P)3/4, and the
running times of the algorithms A are bounded by a
polynomial in the input size and _{e}1/e.
For fixed dimension, one can first compute all vertices of The complexity status of the analogue problem with the polytope
specified by a |