Refined complexity of constraint satisfaction problems

约束满足问题的精细化复杂性

• 批准号: RGPIN-2017-05107
• 负责人: Larose, Benoit
• 金额: $ 1.46万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=656687
• 关键词: Refined; complexity; constraint; satisfaction; problems

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.

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；典型的现实示例包括调度问题、数据库查询、图像处理和频率分配问题。一般来说，确定CSP是否承认一个解决方案是一个算法挑战，但在实践中经常发生约束的形式非常有限，允许使用有效的方法来解决CSP。我们的长期目标是准确分类导致这些易于处理的 CSP 的限制类型。我们的方法基于 CSP 和通用代数之间意想不到且富有成果的联系，这种联系在 90 年代末被发现，并且在过去 20 年中使我们对 CSP 复杂性的理解取得了重大突破。 简而言之，每个约束族都被转化为一个数学对象，其代数属性反映了求解 CSP 的难度。几个精确的猜想已经被提出，预测哪些方程应该导致***在给定的时间和空间限制下可解。该计划的目标是在各种重要的特殊案例中调查并解决这些猜想。对精炼二分猜想的特殊情况的研究必将为完整的 L- 和 NL- 猜想的最终解决方案提供见解。这将使我们对有界宽度的 CSP 的复杂性进行完整的分类，从而对非均匀 CSP 的复杂性有更深入的了解，非均匀 CSP 在计算理论中无处不在，具有从人工智能到数据库理论的广泛应用。