The complexity of valued constraints

这个提议是一个合作申请，涉及到牛津大学的Peter Jeavons教授，伦敦大学皇家霍洛威的大卫科恩教授和法国图卢兹三大学的马丁库珀博士。我们正在寻求资金来扩展和发展一个新的代数复杂性理论。约束满足问题出现在许多实际问题中，例如调度和电路布局，因此这类问题在计算机科学中得到了广泛的研究。所有已知的算法的最一般形式的问题需要指数时间，因此是不切实际的大型案件。然而，已经确定了几个限制，这是足以使限制形式的问题有效地解决。事实上，对问题的仔细数学分析表明，任何特定约束满足问题的计算难度与约束的某些代数性质密切相关。在这个研究项目中，我们正在寻求开发一种新的代数方法，甚至更广泛的一类问题，涉及约束满意度和优化。这样的问题被称为值约束问题。我们希望表明，通过使用一般的代数方法，我们可以识别所有类型的值的约束，可以有效地优化。我们还计划实施的技术，我们开发的新的软件工具，可用于分析任何给定的例子的一个值的约束问题，并解决它使用特殊目的的有效方法时，这些是适用的。

This proposal is a collaborative application involving Professor Peter Jeavons at the University of Oxford, Professor David Cohen at Royal Holloway, University of London, and Dr Martin Cooper at the University of Toulouse III, France.We are seeking funding to extend and develop a novel algebraic theory of complexity for valued constraint satisfaction problems.Constraint satisfaction problems arise in many practical problems, such as scheduling and circuit layout, so this family of problems has been widely studied in computer science. All known algorithms for the most general form of the problem require exponential time, and are therefore impractical for large cases. However, several restrictions have been identified which are sufficient to make the restricted form of the problem efficiently solvable. In fact, a careful mathematical analysis of the problem has shown that the computational difficulty of any particular constraint satisfaction problem is closely related to certain algebraic properties of the constraints. In this research project we are seeking to develop a new algebraic approach to an even wider class of problems which involve both constraint satisfaction and optimisation. Such problems are called valued constraint problems. We hope to show that by using general algebraic methods we can identify all types of valued constraints which can be efficiently optimised. We also plan to implement the techniques we develop in new software tools which can be use to analyse any given example of a valued constraint problem, and solve it using special-purpose efficient methods when these are applicable.