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12 Volume

Input: Polytope P in H-description
Output: The volume of P
Status (general): #P-hard, FPRAS
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 (Aε)ε>0 of randomized algorithms, where, for each ε> 0, Aε computes a number Vε with the property that the probability of (1-ε) vol(P) ≤ Vε ≤ (1+ε) vol(P) is at least 3/4, and the running times of the algorithms Aε are bounded by a polynomial in the input size and 1/ε.

For fixed dimension, one can first compute all vertices of P (see Problem 1) and its face lattice (see Problem 5) both in polynomial time. Then one can construct some triangulation (see Problem 11) of P (e.g., its barycentric subdivision) in polynomial time and compute the volume of P as the sum of the volumes of the (maximal) simplices in the triangulation.

The complexity status of the analogue problem with the polytope specified by a V-description is the same.

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