Applications of algebra and topology to constraint satisfaction problems

在约束满足问题(CSP)中，必须为必须服从各种约束的变量赋值；典型的现实世界示例包括调度问题和数据库查询。一般来说，确定CSP是否接受解决方案是一个算法挑战，但在实践中经常发生的情况是，约束的形式非常有限，允许使用有效的方法来求解CSP。我们的长期目标是准确地对导致这些易处理的CSP的限制进行分类。我们的方法是基于CSP和泛代数之间的一种意想不到的和卓有成效的联系，该联系是由P.Jeavons在90年代末发现的。简而言之，每一族约束都被转化为一个数学对象，其代数性质以某种方式反映了求解CSP的难度。进一步借用了代数拓扑学领域的方法来分析这些代数对象。在过去的5年里，这种代数方法在约束满足问题的算法复杂性的研究中取得了一些重大突破。

In a constraint satisfaction problem (CSP), one must assign values to variables that must obey various constraints; typical real world examples include scheduling problems and database queries. In general, determining whether a CSP admits a solution is an algorithmic challenge, but it often happens in practice that the constraints are of a very restricted form, allowing the use of efficient methods to solve the CSP. Our long-term goal is to classify precisely what kinds of restrictions lead to these tractable CSP's. Our approach is based on an unexpected and fruitful connection between CSP's and universal algebra that was uncovered in the late 90's by P. Jeavons. In short, every family of constraints is transformed into a mathematical object whose algebraic properties somehow reflect the difficulty of solving the CSP. Further methods borrowed from the field of algebraic topology are used to analyse these algebraic objects. This algebraic approach has led to some major breakthroughs in the last 5 years in the study of the algorithmic complexity of constraint satisfaction problems.