具有清晰和保守值模板的混合VCSP
Hybrid VCSPs with crisp and conservative valued templates
摘要：
约束满足问题（CSP）是计算两个关系结构之间的同态${\bf R} \rightarrow {\bf Γ}$的问题。分析其复杂性已经成为一个非常有成果的研究方向，特别是对于固定模板CSP，记为$CSP({\bf Γ})$，其中右侧结构${\bf Γ}$是固定的，而左侧结构${\bf R}$是无约束的.最近，混合设置（写作$CSP_{\mathcal{H}}({\bf Γ})$）引起了一些关注，其中双方同时受到限制。它假设${\bf R}$取自一类关系结构$\mathcal{H}$，另外在逆同态下是封闭的。最后一个属性允许利用已开发的固定模板CSP的代数工具。连接混合CSP和固定模板CSP的关键概念是所谓的“提升语言”。也就是说，这是一种约束语言${\bf Γ}_{\bf R}$，它可以从输入${\bf R}$构造。该语言对任何输入${\bf R}\in\mathcal{H}$的易处理性是混合问题易处理性的必要条件。在第一部分中，我们调查模板${\bf Γ}$，后者的条件不仅是必要的，而且是充分的。我们称这样的模板为${\bf Γ}$广泛易处理的。为此，我们从${\bf Γ}$构造了一个新的有限关系结构${\bf Γ}'$，并将$\mathcal{H}_0$定义为与${\bf Γ}'$同态的一类结构。我们证明了宽可处理性等价于$CSP_{\mathcal{H}_0}({\bf Γ})$的可处理性。我们的证明是基于关键的观察${\bf R}$是同态${\bf Γ}'$当且仅当核心的${\bf Γ}_{\bf R}$是由Siggers多态性保存。类似的结果显示值保守CSP。
英文摘要：
A constraint satisfaction problem (CSP) is a problem of computing a homomorphism ${\bf R} \rightarrow {\bf Γ}$ between two relational structures. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs, denoted $CSP({\bf Γ})$, in which the right side structure ${\bf Γ}$ is fixed and the left side structure ${\bf R}$ is unconstrained. Recently, the hybrid setting, written $CSP_{\mathcal{H}}({\bf Γ})$, where both sides are restricted simultaneously, attracted some attention. It assumes that ${\bf R}$ is taken from a class of relational structures $\mathcal{H}$ that additionally is closed under inverse homomorphisms. The last property allows to exploit algebraic tools that have been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called "lifted language". Namely, this is a constraint language ${\bf Γ}_{\bf R}$ that can be constructed from an input ${\bf R}$. The tractability of that language for any input ${\bf R}\in\mathcal{H}$ is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates ${\bf Γ}$ for which the latter condition is not only necessary, but also is sufficient. We call such templates ${\bf Γ}$ widely tractable. For this purpose, we construct from ${\bf Γ}$ a new finite relational structure ${\bf Γ}'$ and define $\mathcal{H}_0$ as a class of structures homomorphic to ${\bf Γ}'$. We prove that wide tractability is equivalent to the tractability of $CSP_{\mathcal{H}_0}({\bf Γ})$. Our proof is based on the key observation that ${\bf R}$ is homomorphic to ${\bf Γ}'$ if and only if the core of ${\bf Γ}_{\bf R}$ is preserved by a Siggers polymorphism. Analogous result is shown for valued conservative CSPs.
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