We show that CSP is fixed-parameter tractable when parameterized by the
treewidth of a backdoor into any tractable CSP problem over a finite constraint
language. This result combines the two prominent approaches for achieving
tractability for CSP: (i) by structural restrictions on the interaction between
the variables and the constraints and (ii) by language restrictions on the
relations that can be used inside the constraints. Apart from defining the
notion of backdoor-treewidth and showing how backdoors of small treewidth can
be used to efficiently solve CSP, our main technical contribution is a
fixed-parameter algorithm that finds a backdoor of small treewidth.

本研究论文表明，当以任何可转化为有限约束语言的可解 CSP 问题作为背景门限时，CSP 是固定参数可解的，这将结合了两个解决 CSP 可解性问题的突出方法：(i) 通过变量和约束之间交互的结构限制和 (ii) 通过内部约束所使用的关系的语言限制。除了定义背景门限的概念并展示如何利用小的背景门限来高效求解 CSP 之外，本文主要技术贡献是提出了一种寻找小背景门限的固定参数算法。

结合树宽和后门求解 CSP 问题

Combining Treewidth and Backdoors for CSP

In the context of CSPs, a strong backdoor is a subset of variables such that
every complete assignment yields a residual instance guaranteed to have a
specified property. If the property allows efficient solving, then a small
strong backdoor provides a reasonable decomposition of the original instance
into easy instances. An important challenge is the design of algorithms that
can find quickly a small strong backdoor if one exists. We present a systematic
study of the parameterized complexity of backdoor detection when the target
property is a restricted type of constraint language defined by means of a
family of polymorphisms. In particular, we show that under the weak assumption
that the polymorphisms are idempotent, the problem is unlikely to be FPT when
the parameter is either r (the constraint arity) or k (the size of the
backdoor) unless P = NP or FPT = W[2]. When the parameter is k+r, however, we
are able to identify large classes of languages for which the problem of
finding a small backdoor is FPT.

本文对基于约束满足问题的多项式算法强后门检测问题进行了系统研究，特别是当目标属性是由多项式函数族定义的特定约束语言时，我们表明在多项式函数族是幂等的假设下，当参数为 r（约束元数）或 k（后门大小）时，问题不可能是 FPT，除非 P = NP 或 FPT = W [2]。但是当参数为 k + r 时，我们能够识别出大量语言，以便找到小后门的问题是 FPT。

可追溯的约束语言的后门问题

On Backdoors To Tractable Constraint Languages

Lifted probabilistic inference algorithms exploit regularities in the
structure of graphical models to perform inference more efficiently. More
specifically, they identify groups of interchangeable variables and perform
inference once per group, as opposed to once per variable. The groups are
defined by means of constraints, so the flexibility of the grouping is
determined by the expressivity of the constraint language. Existing approaches
for exact lifted inference use specific languages for (in)equality constraints,
which often have limited expressivity. In this article, we decouple lifted
inference from the constraint language. We define operators for lifted
inference in terms of relational algebra operators, so that they operate on the
semantic level (the constraints extension) rather than on the syntactic level,
making them language-independent. As a result, lifted inference can be
performed using more powerful constraint languages, which provide more
opportunities for lifting. We empirically demonstrate that this can improve
inference efficiency by orders of magnitude, allowing exact inference where
until now only approximate inference was feasible.

该文章将提升的推断与约束语言解耦，并通过关系代数运算符定义提升的推断算子，使它们能够在语义层面（约束扩展）上进行操作，从而使其与语言无关。通过使用更强有力的约束语言，飞跃推断的效率得到了数量级的提升，从而允许进行精确推断，而不是仅使用近似推断。