The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through either value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m-wBTP allows us to merge more values than BTP. DBTP, ∀∃-BTP, k-BTP, WBTP and m-wBTP permit us to capture more tractable instances than BTP. However, except BTP, none of these extensions allows variable elimination while preserving satisfiability. Moreover, k-BTP and m-wBTP define bigger tractable classes around BTP but both of them generally need a high level of consistency. Here, we introduce a new weaker form of BTP, called m-fBTP for flexible broken-triangle property, which will represent a compromise between most of these previous tractable properties based on BTP. m-fBTP allows us on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define a new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but fewer than m-wBTP. The binary CSP instances satisfying m-fBTP are solved by algorithms of the state-of-the-art like MAC and RFL in polynomial time. An open question is whether it is possible to compute, in polynomial time, the existence of some variable ordering for which a given instance satisfies m-fBTP.

中文翻译：

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关于针对二进制CSP的BTP的新扩展

通过在多项式时间内解决约束满足问题（CSP）并通过值合并或变量消除来减小搜索空间大小，对三角形的研究变得越来越雄心勃勃。为了扩展捕获的易处理CSP实例的数量和/或合并值的数量，在扩展此重要概念（例如双破三角形和弱破三角形）方面已经取得了可观的进展。具体而言，m -wBTP允许我们合并比BTP更多的值。DBTP，∀∃-BTP，k -BTP，WBTP和m- wBTP允许我们捕获比BTP更加容易处理的实例。但是，除了BTP之外，这些扩展都不能在保持可满足性的同时消除变量。此外，k -BTP和m -wBTP在BTP周围定义了较大的易处理类，但是它们通常都需要高度的一致性。在这里，我们介绍了一种新的较弱的BTP形式，称为m -fBTP，以实现灵活的破碎三角形属性，这将代表大多数基于BTP的先前易处理属性之间的折衷。m -fBTP一方面使我们可以在保持可满足性的同时比BTP消除更多的变量，另一方面使我们可以定义一个更大的易处理类，对于该类来说，电弧一致性是一个决策过程。同样，m -fBTP允许合并的值比BTP多，但少于m -wBTP。满足m的二进制CSP实例-fBTP通过多项式时间内的最新算法（例如MAC和RFL）解决。一个悬而未决的问题是，是否可以在多项式时间内计算给定实例满足m -fBTP的某些变量排序的存在。