Applications of algebraic methods in combinatorial problems

代数方法在组合问题中的应用

• 批准号: RGPIN-2020-05481
• 负责人: Bulatov, Andrei
• 金额: $ 4.66万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750547
• 关键词: Applications; algebraic; methods; combinatorial; problems

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 including many crucial real-world problems in scheduling, routing, databases, etc. 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 focuses on the theoretical side of the CSP research. One of its main goals is to precisely understand which CSPs and counting CSPs can be solved efficiently and design efficient solution algorithms. Major progress has been made very recently in understanding the hardness of the CSP, and a natural next step is to extend the methods used to achieve this breakthrough to a wider range of problems. One of the most efficient tools in the study of the CSP applies methods of abstract algebra. We plan to develop these methods to work in other areas. One of such areas is approximate counting. This area has surprising connections to statistical physics, where such numbers affect the properties of large ensembles of particles, and, in particular, help to locate phase transition thresholds, at which, for instance, liquid turns to gas. While the CSP and the counting CSP are well established research areas, we also plan to venture into very new territory. A further generalization of the CSP, the Promise CSP, has been recently proposed, in which, given two CSP instances, a more restrictive one, and a more relaxed one, we are promised that the restrictive instance has a solution and asked to find a solution of the relaxed instance. This problem has much greater expressive power than the regular CSP. Certain elements of the algebraic approach to the Promise CSP have been developed recently, and a project aiming at a classification of Promise CSPs is very active.

约束满意度问题（CSP）提供了一个一般框架，在这种框架中，可以自然地表达各种各样的问题，包括在调度，路由，数据库等方面的许多至关重要的现实世界问题。CSP中的目的是找到对给定的变量的值分配给定变量的主体，可以同时分配特定的变量。在计数约束满意度问题中，目标是找到此类作业的数量。因此，CSP在理论和实用的许多应用中都非常重要。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，改善现有算法，设计更一般和统一的算法，并找到证据表明某些问题在一般情况下无法有效解决。在实际应用中，限制和满足性求解器现在是从计划到硬件验证的广泛领域中的标准工具。研究CSP的进步将加快此类求解器并扩大其适用性。 该项目着重于CSP研究的理论方面。它的主要目标之一是精确了解哪些CSP和计数CSP可以有效地解决并设计有效的解决方案算法。最近在理解CSP的硬度方面取得了重大进展，自然的下一步是将实现这一突破的方法扩展到更广泛的问题。 CSP研究中最有效的工具之一是采用抽象代数的方法。我们计划开发这些方法以在其他领域工作。这样的领域之一是近似计数。该区域与统计物理学有令人惊讶的连接，其中这样的数字会影响颗粒的大型颗粒的特性，尤其是有助于定位相变阈值，例如，液体转向气体。尽管CSP和计数CSP是良好的研究领域，但我们还计划冒险进入非常新的领域。最近提出了对CSP的进一步概括，即Promise CSP，其中，给定两个CSP实例，一个更限制的实例，并且更放松的情况，我们承诺，限制性实例具有解决方案，并要求找到放松实例的解决方案。与常规CSP相比，此问题具有更大的表达能力。代数方法的某些要素最近已经开发出来，并且针对Promise CSP的分类的项目非常活跃。