From a theoretical viewpoint, the (tree-)decomposition methods offer a good approach for solving Constraint Satisfaction Problems (CSPs) when their (tree)-width is small. In this case, they have often shown their practical interest. So, the literature (coming from Mathematics, OR or AI) has concentrated its efforts on the minimization of a single parameter, namely the tree-width. Nevertheless, experimental studies have shown that this parameter is not always the most relevant to consider when solving CSPs. So, in this paper, we highlight two fundamental problems related to the use of tree-decomposition and for which we offer two particularly appropriate solutions. First, we experimentally show that the decomposition algorithms of the state of the art produce clusters (a tree-decomposition is a rooted tree of clusters) having several connected components. We highlight the fact that such clusters create a real disadvantage which affects significantly the efficiency of solving methods. To avoid this problem, we consider here a new graph decomposition called Bag-Connected Tree-Decomposition, which considers only tree-decompositions such that each cluster is connected. We analyze such decompositions from an algorithmic point of view, especially in order to propose a first polynomial time algorithm to compute them. Moreover, even if we consider a very well suited decomposition, it is well known that sometimes, a bad choice for the root cluster may significantly degrade the performance of the solving. We highlight an explanation of this degradation and we propose a solution based on restart techniques. Then, we present a new version of the BTD algorithm (for Backtracking with Tree-Decomposition Jégou and Terrioux, Artificial Intelligence, 146 43–75 28) integrating restart techniques. From a theoretical viewpoint, we prove that reduced nld-nogoods can be safely recorded during the search and that their size is smaller than ones recorded by MAC+RST+NG (Lecoutre et al., JSAT, 1(3–4) 147–167 34). We also show how structural (no)goods may be exploited when the search restarts from a new root cluster. Finally, from a practical viewpoint, we show experimentally the benefits of using independently bag-connected tree-decompositions and restart techniques for solving CSPs by decomposition methods. Above all, we experimentally highlight the advantages brought by exploiting jointly these improvements in order to respond to two major problems generally encountered when solving CSPs by decomposition methods.

中文翻译：

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结合重启，杂物和袋装分解来解决CSP

从理论上讲，（树）分解方法为解决（树）宽度小的约束满足问题（CSP）提供了一种很好的方法。在这种情况下，他们经常表现出实际的兴趣。因此，文献（来自数学，OR或AI）将精力集中在最小化单个参数（即树宽）上。尽管如此，实验研究表明，解决CSP时，该参数并非始终是最相关的。因此，在本文中，我们重点介绍了与树分解的使用相关的两个基本问题，为此我们提供了两个特别合适的解决方案。第一，我们通过实验证明，现有技术的分解算法会产生具有多个连接组件的聚类（树分解是聚类的根树）。我们强调了这样一个事实，即这样的簇会产生真正的劣势，这会严重影响求解方法的效率。为了避免这个问题，我们在这里考虑一个称为Bag-Connected Tree-Decomposition，它仅考虑树分解，以便每个集群都已连接。我们从算法的角度分析这种分解，特别是为了提出第一个多项式时间算法来计算它们。而且，即使我们认为分解非常合适，也众所周知，有时根簇的错误选择可能会大大降低求解的性能。我们重点介绍了这种性能下降的原因，并提出了一种基于重启技术的解决方案。然后，我们提出了BTD算法的新版本（用于通过树分解Jégou和Terrioux进行回溯，人工智能146）43–75 28）整合重启技术。从理论上讲，我们证明了减少的nld-nogoods可以在搜索过程中安全记录，并且其大小小于MAC + RST + NG记录的数量（Lecoutre等人，JSAT，1（3-4） 147 167 34）。我们还展示了当搜索从新的根群集重新开始时，如何利用结构性（无）商品。最后，从实际角度出发，我们通过实验展示了使用独立的袋连接树分解和重新启动技术通过分解方法求解CSP的好处。最重要的是，我们实验性地强调了联合利用这些改进所带来的优势，以应对通过分解方法求解CSP时通常遇到的两个主要问题。