The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

2026-09-10

AI Summary

本文通过引入舒尔定理和SL(2,C)的有限子群分类等新方法，解决了4正则图上具有复数值4元签名的Holant问题的计算复杂性二分法。

Abstract

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

Problem

• Holant problems
• computational complexity
• 4-regular graphs
• complex-valued signatures

Innovation

• Schur's theorem
• finite subgroups of SL(2, C)
• stable subgroup sequences