Approximation Algorithms for NP-hard Optimization Problems

NP 难优化问题的近似算法

• 批准号: RGPIN-2014-06302
• 负责人: Gaur, Daya
• 金额: $ 1.46万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567720
• 关键词: Approximation; Algorithms; NP; hard; Optimization

It is generally believed that efficient algorithms do not exist for finding an optimal solution to NP-hard opti- mization problems. A natural way to deal with the inability to find exact solutions efficiently is to trade the quality of solution for the computation time. Approximation algorithms do precisely that. Approximation algorithms not only provide an approximate solution in polynomial time, they also provide a certificate of optimality for the solution. One can always refine and tune an approximation algorithm to specific class of instances arising in practice, thereby improving the performance ratio. Primary goal of this research is to further the theory and praxis of the design of approximation algorithms. We will design approximation algorithms for problems arising in the bioinformatics, facility location, schedul- ing, and machine learning domains. Our approach for designing approximation algorithms has theoretical underpinnings in combinatorial algorithms, linear and semi-definite programming. Linear programming based approaches have enjoyed a great deal of success in the recent past. The basic framework entails de- scribing the optimization problem as an integer linear program or a non-linear program over integer variables. A suitable linear programming relaxation or a semi-definite programming relaxation is constructed. The re- laxation is solved using efficient algorithms. A fractional solution thus obtained is converted to an integral solution using some scheme. Care is taken to ensure that the process of converting the fractional solution to an integral solution does not increase the cost of the solution too much. Another approach is to simultane- ously construct an integral primal solution and candidate dual solution (possibly fractional) iteratively using the primal-dual schema. Primal-dual schema has the advantage that one can work with an exponential sized formulation without having to resort to a separation oracle. Approaches based on the primal-dual schema have been very successful for certain types of optimization problems. Integrality gap of an integer program- ming formulation is the gap in the cost of the optimal fractional and the cost of the optimal integral solution. Approximation algorithms based on linear or semi-definite programming have performance ratio no better than the integrality gap of the relaxation. Therefore integer programs with small integrality gap are critical to the success of such approximation algorithms. The immediate program is i) to develop relaxations with bounded integrality gap for the optimization problems of interest or show none exists, ii) to develop combi- natorial algorithms for solving the relaxations where possible, and iii) to develop provably good strategies for converting the fractional solutions to the relaxations to integral solution. There has been considerable research activity on the hardness of approximations in the last few years and several deep results on the hard- ness of approximations have been obtained. The focus of this research is on the design of approximation algorithms and we will draw on the results on hardness of approximations to guide the program. On the theoretical front we will push the frontier in approximation algorithms design. Research conducted as part of this project will have prospect for commercialization and will be of immediate interest to the industry. Knowledge generated will be protected using patents (where applicable) and disseminated in high quality journals and conferences. The program will produce highly skilled manpower; skilled in the use of discrete optimization theory and tools.

人们普遍认为，不存在找到 NP 难优化问题最优解的有效算法。解决无法有效找到精确解决方案的自然方法是用解决方案的质量来换取计算时间。近似算法正是这样做的。近似算法不仅提供多项式时间内的近似解，还提供解的最优性证书。人们总是可以针对实践中出现的特定实例类别来完善和调整近似算法，从而提高性能比。本研究的主要目标是进一步推进近似算法设计的理论和实践。我们将为生物信息学、设施定位、调度和机器学习领域中出现的问题设计近似算法。我们设计近似算法的方法具有组合算法、线性和半定规划的理论基础。基于线性规划的方法近年来取得了巨大的成功。基本框架需要将优化问题描述为整数线性程序或整数变量上的非线性程序。构造合适的线性规划松弛或半定规划松弛。使用有效的算法来解决松弛问题。使用某种方案将由此获得的分数解转换为积分解。要注意确保将分数解转换为积分解的过程不会过多地增加解的成本。另一种方法是使用原始对偶模式迭代地同时构建积分原始解和候选对偶解（可能是分数）。原始对偶模式的优点是可以使用指数大小的公式，而无需求助于分离预言机。基于原对偶模式的方法对于某些类型的优化问题非常成功。整数规划公式的积分差距是最优分数的成本与最优积分解的成本之间的差距。基于线性或半定规划的逼近算法的性能比并不优于松弛的完整性差距。因此，具有较小完整性差距的整数规划对于此类近似算法的成功至关重要。直接的计划是 i) 为感兴趣的优化问题开发具有有界完整性差距的松弛或表明不存在，ii) 开发组合算法来尽可能解决松弛，以及 iii) 开发可证明的良好策略来转换积分解松弛的分数解。在过去的几年里，关于近似硬度的研究活动非常多，并且已经获得了一些关于近似硬度的深入成果。本研究的重点是近似算法的设计，我们将利用近似硬度的结果来指导程序。在理论方面，我们将推动近似算法设计的前沿。作为该项目一部分进行的研究将具有商业化前景，并将引起业界的直接兴趣。产生的知识将使用专利（如适用）进行保护，并在高质量期刊和会议上传播。该计划将培养高技能人才；熟练运用离散优化理论和工具。