Approximate counting of vertices of 0/1 polytopes: a stronger hardness result

Heng Guo; Mark Jerrum

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arXiv · 2608.22290

Approximate counting of vertices of 0/1 polytopes: a stronger hardness result

Heng Guo·Mark Jerrum

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Abstract

We show that approximately counting the vertices of a bounded 0/1 polytope, presented as a system of rational linear inequalities, is, informally speaking, NP-hard. In particular, there is no FPRAS for this problem unless RP=NP. The proof is by a reduction from approximately counting homomorphisms from a given graph to a particular four-vertex graph. The main proof ideas were found using GPT-5.6 Sol Ultra.

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Heng Guo, Mark Jerrum. 2026-08-23. Approximate counting of vertices of 0/1 polytopes: a stronger hardness result. https://arxiv.org/abs/2608.22290

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