The complexity of the constraint satisfaction problem and its variants

约束满足问题（CSP）提供了一个通用的框架，在其中可以以自然的方式表达各种各样的问题。CSP的目标是找到一组给定变量的值的分配，这些值可以同时分配给某些指定的变量子集;在计数约束满足问题中，目标是找到这样的分配的数量。因此，CSP在许多应用中非常重要，无论是理论还是实践。在理论应用中，我们寻找新的算法思想，可以帮助解决以前无法解决的问题，改进现有的算法，设计更通用和统一的算法，并找到证据，证明某些问题在一般情况下无法有效解决。在实际应用中，约束和可满足性求解器现在是从调度到硬件验证的广泛领域中的标准工具。CSP研究的进展将加快此类求解器的速度，并扩大其适用范围。这个项目将有助于在约束问题的两个方面，虽然它的主要重点是在理论方面。它的主要目标之一是精确地了解哪些CSP可以有效地解决，并设计一个有效的解决方案算法。这是计算机科学中长期存在的开放问题之一，但有证据表明我们可能正在接近它的解决方案。该项目的另一个组成部分是研究计算约束满足问题。这一领域与统计物理学不太可能有联系，在统计物理学中，这些数字会影响大量粒子的性质，特别是有助于确定相变阈值，例如，液体变成气体。最后，本计画提出在比标准最坏情形方法更接近实际情况的架构下，研究效率并设计新的演算法。众所周知，实际问题往往具有某种称为幂律分布的性质，我们计划在数学上研究满足此性质的CSP实例上的各种算法的行为。

The constraint satisfaction problem (CSP) provides a general framework in which it is possible to express, in a natural way, a wide variety of problems. The aim in a CSP is to find an assignment of values to a given set of variables, subject to constraints on the values which can be simultaneously assigned to certain specified subsets of variables; in the counting constraint satisfaction problem the objective is to find the number of such assignments. The CSP is therefore very important in a number of applications, both theoretical and practical. In theoretical applications we look for new algorithmic ideas that can help to solve problems that could not be solved before, to improve the existing algorithms, to design more general and uniform algorithms, and to find evidence that certain problems are not efficiently solvable in the general case. In practical applications, constraint and satisfiability solvers are now standard tools in a wide range of areas from scheduling to hardware verification. Advances in the study of the CSP will speed up such solvers and expand their area of applicability.*** This project will contribute in both aspects of constraint problems, although its main focus is on the theoretical side. One of its main goals is to precisely understand which CSPs can be solved efficiently and design an efficient solution algorithm. This is one of the long standing open problems in computer science, but there is evidence that we may be approaching its resolution. Another component of the project is the study of counting constraint satisfaction problems. This area has unlikely connections to statistical physics, where such numbers affect the properties of large ensembles of particle, and, in particular,  help to locate phase transition thresholds, at which, for instance, liquid turns to gas. Finally, this project proposes to study the efficiency and to design new algorithms in the framework that is closer to practical circumstances than the standard worst case approach. It is known that practical problems tend to have a certain property called the power law distribution, and we plan to mathematically study the behaviour of various algorithms on instance of the CSP satisfying this property.**