Approximation Algorithms for Combinatorial Optimization Problems

在制造、运输、通信、金融和许多其他领域中的许多优化问题需要优化可用资源使用的解决方案。这些问题的一个重要类别是组合优化问题。这些问题中的许多问题太复杂，如果不使用计算机就无法解决，因此需要有效的计算机算法来解决它们。然而，有强有力的理论证据表明，即使是现代快速的计算机也不足以有效地解决许多这些优化问题;这些问题得到了NP难的技术名称。 尽管似乎不可能有效地解决NP难问题（或者从技术上讲，在多项式时间内），但我们仍然需要处理它们，因为它们出现在工业和商业应用中。处理这些问题的一个重要工具是近似算法;这些是有效的算法，产生的解决方案的值是在一些因素c从最佳。了解近似算法产生的解有多接近最优解，有助于我们确定解决实际相关的复杂问题的能力。 本研究主要针对网路问题、排程问题与排样问题三种问题，设计近似演算法。网络是最基本的建模工具之一，在优化和调度和包装问题是非常重要的领域，需要有效地管理有限的资源。 目标： 1.设计有效的近似算法，利用特殊结构的限制情况下的调度，包装，和网络优化问题，在实际应用中出现。 实际应用的固有性质多次限制了在由它们引起的优化问题的实例中可能出现的输入的集合。我计划研究这些受限实例的结构，为它们发现好的设计技术。 2.阐述设计排程、包装和网路问题近似演算法的新技术，以产生具有实际执行时间的近似演算法。 我将致力于扩展和结合现有的近似算法的设计技术，使它们适用于更大的问题集。 3.研究近似算法产生的解的质量与其运行时间之间的权衡。 在实际应用中，有必要限制算法可以运行的时间量。我计划在理解什么是最好的解决方案，一个人可以希望在给定的时间内获得一个特定的问题。 我希望我的研究，导致近似算法，更适合于网络，调度和包装问题的实际应用比现有的，无论是在近似比和运行时间。本研究具有重要的理论和实践意义。

Many optimization problems in manufacturing, transportation, communications, finances, and many other fields require solutions that optimize the use of available resources. An important class of these problems are combinatorial optimization problems. Many of these problems are too complex to solve without the use of computers and, hence, efficient computer algorithms for solving them are needed. However, there is strong theoretical evidence suggesting that even modern fast computers are not powerful enough to efficiently solve many of these optimization problems; these problems get the technical name of NP-hard. Despite the seeming impossibility for solving NP-hard problems efficiently (or,technically, in polynomial time), we still need to deal with them since they arise in industrial and commercial applications. One important tool for dealing with these problems is approximation algorithms; these are efficient algorithms that yield solutions whose values are within some factor c from the optimum. Knowing how close to optimal are the solutions produced by approximation algorithms helps us determine how well we can solve complex problems of practical relevance. This research focuses on the design of approximation algorithms for three kinds of problems: network, scheduling and packing problems. Networks are one of the most fundamental modelling tools in optimization and scheduling and packing problems are of great importance in domains requiring the effective management of limited resources. Goals: 1. To design efficient approximation algorithms that exploit the special structure of restricted instances of scheduling, packing, and network optimization problems that arise in practical applications. The inherent nature of practical applications many times restricts the set of inputs that my appear in instances of optimization problems arising from them. I plan to work on exploiting the structure of these restricted instances to discover good design techniques for them. 2. To formulate new techniques for the design of approximation algorithms for scheduling, packing, and network problems that yield approximation algorithms with practical running times. I will work on extending and combining existing design techniques for approximation algorithms so they are applicable to a larger set of problems. 3. To study the trade-off between the quality of solutions produced by approximation algorithms and their running times. In practical applications it is necessary to limit the amount of time that an algorithm can run. I plan to work on understanding what is the best solution that one can hope to obtain for a particular problem within a given amount of time. I expect my research to lead to approximation algorithms that are better suited than existing ones for practical applications of network, scheduling and packing problems, both with respect to approximation ratio and running time. This research is of theoretical and practical importance.