Deterministic subexponential-factor approximation of arbitrary polytope volumes

Determine whether the volume of an arbitrary polytope can be approximated to within a subexponential multiplicative factor by a deterministic polynomial-time algorithm.

Background

Randomized polynomial-time algorithms are known to approximate the volume of a polytope to within a (1 ± ε) multiplicative factor. In contrast, the paper explains that deterministic volume approximation remains unresolved. The authors obtain a deterministic exp(O(n log n))-factor approximation for Kostka polytopes under specific restrictions on the partition, but this does not resolve the corresponding problem for arbitrary polytopes.

The unresolved question is stated as a broader complexity-theoretic problem concerning deterministic approximation of polytope volumes, particularly to subexponential factors in the dimension. This provides context for the paper’s contribution, which addresses a structured family of high-dimensional polytopes rather than arbitrary polytopes.