The $\mathcal {H}$-colouring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSPs): an $\mathcal {H}$-colouring of a graph $\mathcal {G}$ is just a homomorphism from $\mathcal {G}$ to $\mathcal {H}$ and the problem is to decide for fixed $\mathcal {H}$, given $\mathcal {G}$, if a homomorphism exists or not. The dichotomy theorem for the $\mathcal {H}$-colouring problem was proved by Hell and Nešetřil (1990, J. Comb. Theory Ser. B, 48, 92–110) (an analogous theorem for all CSPs was recently proved by Zhuk (2020, J. ACM, 67, 1–78) and Bulatov (2017, FOCS, 58, 319–330)), and it says that for each $\mathcal {H}$, the problem is either $p$-time decidable or $NP$-complete. Since negations of unsatisfiable instances of CSP can be expressed as propositional tautologies, it seems to be natural to investigate the proof complexity of CSP. We show that the decision algorithm in the $p$-time case of the $\mathcal {H}$-colouring problem can be formalized in a relatively weak theory and that the tautologies expressing the negative instances for such $\mathcal {H}$ have polynomial proofs in propositional proof system $R^*(log)$, a mild extension of resolution. In fact, when the formulas are expressed as unsatisfiable sets of clauses, they have $p$-size resolution proofs. To establish this, we use a well-known connection between theories of bounded arithmetic and propositional proof systems. This upper bound follows also from a different construction in [1]. We complement this result by a lower bound result that holds for many weak proof systems for a special example of $NP$-complete case of the $\mathcal {H}$-colouring problem, using known results about the proof complexity of the pigeonhole principle. The main goal of our work is to start the development of some of the theories beyond the CSP dichotomy theorem in bounded arithmetic. We aim eventually—in a subsequent work—to formalize in such a theory the soundness of Zhuk’s algorithm, extending the upper bound proved here from undirected simple graphs to the general case of directed graphs in some logical calculi.

中文翻译：

---

ℋ-证明复杂性中的着色二分法

无向简单图的 $\mathcal {H}$-着色问题是一大类约束满足问题 (CSP) 中的一个计算问题：图 $\mathcal {G 的 $\mathcal {H}$-着色}$ 只是从 $\mathcal {G}$ 到 $\mathcal {H}$ 的同态，问题是确定固定 $\mathcal {H}$，给定 $\mathcal {G}$，如果同态存在与否。Hell 和 Nešetřil (1990, J. Comb. Theory Ser. B, 48, 92-110) 证明了 $\mathcal {H}$ 着色问题的二分定理（所有 CSP 的类似定理最近由Zhuk (2020, J. ACM, 67, 1-78) 和 Bulatov (2017, FOCS, 58, 319-330))，它说对于每个 $\mathcal {H}$，问题要么是 $p$ -时间可判定或$NP$-完成。由于 CSP 不可满足实例的否定可以表示为命题重言式，研究 CSP 的证明复杂性似乎很自然。我们证明了在 $p$-time 情况下 $\mathcal {H}$-着色问题的决策算法可以在一个相对较弱的理论中形式化，并且重言式表示这种 $\mathcal {H} 的否定实例$ 在命题证明系统 $R^*(log)$ 中具有多项式证明，这是对分辨率的温和扩展。事实上，当公式表示为不可满足的子句集时，它们具有 $p$-size 分辨率证明。为了建立这一点，我们使用了有界算术理论和命题证明系统之间的一个众所周知的联系。这个上限也来自 [1] 中的不同结构。我们通过一个下界结果来补充这个结果，该结果适用于许多弱证明系统，用于 $\mathcal {H}$-着色问题的 $NP$-完全情况的特殊示例，使用关于鸽笼证明复杂性的已知结果原则。我们工作的主要目标是开始在有界算术中开发一些超越 CSP 二分定理的理论。我们最终的目标是——在随后的工作中——在这样的理论中形式化朱克算法的合理性，将这里证明的上限从无向简单图扩展到一些逻辑演算中有向图的一般情况。