Approximation Algorithms for Combinatorial Optimization Problems

制造业、交通运输、通信、金融和许多其他领域的许多优化问题都需要优化可用资源使用的解决方案。其中一类重要的问题是组合优化问题。这些问题中有许多太复杂，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，即使是现代快速计算机也不够强大，无法有效地解决许多这些优化问题；这些问题被称为NP-hard。尽管看起来不可能有效地解决np困难问题（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；这些都是有效的算法，它们产生的解的值与最优值相差不超过某个因子c。知道近似算法产生的解有多接近最优，有助于我们确定解决实际相关的复杂问题的能力。本文主要研究了网络、调度和包装三种问题的近似算法设计。网络是优化、调度和包装问题中最基本的建模工具之一，在需要有效管理有限资源的领域中具有重要意义。目标:1。设计有效的近似算法，利用在实际应用中出现的调度，包装和网络优化问题的限制实例的特殊结构。实际应用程序的固有性质很多时候限制了在由它们引起的优化问题实例中出现的输入集。我计划利用这些受限实例的结构，为它们发现良好的设计技术。2. 为调度、包装和网络问题的近似算法设计制定新技术，以产生具有实际运行时间的近似算法。我将致力于扩展和结合现有的近似算法设计技术，使它们适用于更大的问题集。3. 研究由近似算法产生的解的质量和它们的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。我打算努力理解在给定的时间内，一个人可以希望获得一个特定问题的最佳解决方案是什么。我希望我的研究能够产生比现有的近似算法更适合网络、调度和包装问题的实际应用，无论是在近似比还是运行时间方面。本研究具有一定的理论和实践意义。

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.