混合CSP结构约束的有效性
Effectiveness of Structural Restrictions for Hybrid CSPs
摘要：
约束满足问题（CSP）是计算机科学中的一个基本算法问题。它可以等价地表述为计算两个关系结构之间的同态$\mbox{$ \bR \rightarrow \bGamma $}$，例如\在两个有向图之间。分析其复杂性一直是一个突出的研究方向，特别是对于{ \em固定模板CSP}，其中右侧$\bGamma$是固定的，左侧$\bR$是无约束的。同时限制两侧的{ \em hybrid}设置的已知结果要少得多。它假设$\bR$属于某一类关系结构（本文称为{ \em结构限制}）。我们研究哪些结构限制是{ \em有效}的，即存在固定的模板$\bGamma$（来自某类语言），当$\bR$受到限制时，该问题是可处理的，否则是NP难的。我们提供了一个表征结构的限制，是{ \em下逆同态封闭}。该判据是基于本文定义的关系结构的{ \em色数}，它推广了图的标准色数。作为我们的主要工具，我们使用的代数机械开发的固定模板CSP。为了将其应用到我们的案例中，我们引入了一个新的结构，称为“提升语言”。“我们还给出了对应于小闭图族的结构限制的特征，将结果扩展到某些值CSP（即保守值语言），以及具有有序变量的CSP，具有大围长的结构上的（值）CSP，以及某些受限图族（包括具有大围长的图）上的最大权独立集问题的状态含义。
英文摘要：
Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism $\mbox{$\bR \rightarrow \bGamma$}$ between two relational structures, e.g.\ between two directed graphs. Analyzing its complexity has been a prominent research direction, especially for {\em fixed template CSPs} in which the right side $\bGamma$ is fixed and the left side $\bR$ is unconstrained. Far fewer results are known for the {\em hybrid} setting that restricts both sides simultaneously. It assumes that $\bR$ belongs to a certain class of relational structures (called a {\em structural restriction} in this paper). We study which structural restrictions are {\em effective}, i.e.\ there exists a fixed template $\bGamma$ (from a certain class of languages) for which the problem is tractable when $\bR$ is restricted, and NP-hard otherwise. We provide a characterization for structural restrictions that are {\em closed under inverse homomorphisms}. The criterion is based on the {\em chromatic number} of a relational structure defined in this paper; it generalizes the standard chromatic number of a graph. As our main tool, we use the algebraic machinery developed for fixed template CSPs. To apply it to our case, we introduce a new construction called a "lifted language." We also give a characterization for structural restrictions corresponding to minor-closed families of graphs, extend results to certain Valued CSPs (namely conservative valued languages), and state implications for CSPs with ordered variables, (valued) CSPs on structures with large girth, and for the maximum weight independent set problem on some restricted families of graphs including graphs with large girth.
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