Refined complexity of constraint satisfaction problems

在约束满足问题（CSP）中，必须为必须满足各种约束的变量赋值;典型的真实的例子包括调度问题，数据库查询，图像处理和频率分配问题。一般来说，确定CSP是否承认解决方案是一个算法挑战，但在实践中经常发生的是，约束是一个非常有限的形式，允许使用有效的方法来解决CSP。我们的长期目标是准确地分类哪些类型的限制导致这些易处理的CSP。我们的方法是基于一个意想不到的和富有成效的连接CSP的和通用代数是在90年代末发现的，并导致了重大突破，在我们的理解的复杂性CSP在过去的20年。 简而言之，每一个家庭的约束转换成一个数学对象，其代数属性反映了解决CSP的难度。已经制定了几个精确的公式，预测哪些方程应该导致 在给定的时间和空间限制下的可解性。该计划的目标是在各种重要的特殊情况下调查和解决这些问题。对精化二分法的特殊情况的研究必将为完全L-和NL-二分法的最终解提供深入的见解。这将给我们一个完整的分类有限宽度的CSP的复杂性，从而更深入地了解非均匀CSP的复杂性，这是无处不在的计算理论，从人工智能到数据库理论的广泛应用。

In a constraint satisfaction problem (CSP), one must assign values to variables that must satisfy various constraints; typical real world examples include scheduling problems, database queries, image-processing, and frequency assignment problems. 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, and which has led to major breakthroughs in our understanding of the complexity of CSPs over the past 20 years. In short, every family of constraints is transformed into a mathematical object whose algebraic properties reflect the difficulty of solving the CSP. Several precise conjectures have been formulated, predicting which equations should lead to solvability with given time and space restrictions. The goal of this program is to investigate and solve these conjectures in various important special cases. The investigation of special cases of the refined dichotomy conjectures is bound to provide insights into an eventual solution of the full L- and NL- conjectures. This will give us a complete classification of the complexity of CSPs of bounded width, and hence a much deeper understanding of the complexity of non-uniform CSPs, which are ubiquitous in the theory of computing, with wide-ranging applications from artificial intelligence to database theory.