AF: Small: Mathematical Programming Methods in Approximation

数学规划技术，如线性规划和semi；夜间编程有&amp；#64257；Rmly在近似算法设计工具包中确立了自己的价值。它们常被用作难以处理的组合优化问题的松弛，并作为获得近似算法的设计指南。这个项目试图提高我们对数学规划技术在几个基本优化问题上的优势和劣势的理解，并提出研究一般方法来加强我们目前已知的数学规划技术。这个项目的主要目标如下：(a)尝试为独特的游戏设计更好的算法？一个约束满足问题，已知捕获当前半导体的局限性。在近似算法中使用的Nite编程方法。打败当前最好的算法将必然涉及开发新技术，克服当前SDP方法的局限性。(b)了解通过提升和项目程序获得的强化松弛的结构，并开发利用这些更强松弛提供的附加信息的技术，以获得更好的近似算法。(c)努力缩小我们对基本优化问题的近似性的理解上的巨大差距。成功实现项目目标将需要重要的因素。近似值算法的最新进展。这项研究可能会开发出广泛适用于几个优化问题的工具和技术。将根据本项目的研究成果以及该领域的新进展，编写研究生和本科生课程的教材。

Mathematical programming techniques like linear programming and semidefinite programming have firmly established themselves as valuable tools in the approximation algorithm design toolkit. They are often used as tractable relaxations to hard combinatorial optimization problems and as design guides to obtain approximation algorithms. This project attempts to enhance our understanding of the strengths and weaknesses of mathematical programming techniques for several fundamental optimization problems and proposes to investigate general methods to strengthen our currently known mathematical programming techniques.The broad goals of this project are the following: (a) Attempt to devise better algorithms for unique games ? a constraint satisfaction problem that is known to capture the limitations of current semidefinite programming methods used in approximation algorithms. Beating the current best algorithms will necessarily involve developing new techniques that overcome the limitations of current SDP approaches. (b) Understand the structure of strengthened relaxations obtained by lift-and-project procedures and develop techniques to exploit the additional information provided by these stronger relaxations to obtain better approximation algorithms. (c) Work towards closing large gaps in our understanding of the approximability of fundamental optimization problems.Successfully achieving the project goals will entail significant advances in the state of the art for approximation algorithms. The research could potentially develop tools and techniques with broad applicability to several optimization problems. Course materials for graduate and undergraduate courses will be developed distilling research results of this project, as well as new developments in the field.