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

2026-08-23

AI Summary

本文证明了近似计数0/1多面体顶点问题的NP难性，通过将图同态问题归约到该问题上，除非RP=NP否则不存在FPRAS解法。

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.

Problem

• approximate counting
• 0/1 polytopes
• NP-hard
• FPRAS

Innovation

• approximate counting
• 0/1 polytope
• NP-hard
• FPRAS
• graph homomorphisms