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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634849
• 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 thequality of solution for the computation time. Approximation algorithms do precisely that. Approximationalgorithms not only provide an approximate solution in polynomial time, they also provide a certificate ofoptimality for the solution. One can always refine and tune an approximation algorithm to specific class ofinstances 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. Wewill design approximation algorithms for problems arising in the bioinformatics, facility location, schedul-ing, and machine learning domains. Our approach for designing approximation algorithms has theoreticalunderpinnings in combinatorial algorithms, linear and semi-definite programming. Linear programmingbased 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 integralsolution using some scheme. Care is taken to ensure that the process of converting the fractional solution toan 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 usingthe primal-dual schema. Primal-dual schema has the advantage that one can work with an exponential sizedformulation without having to resort to a separation oracle. Approaches based on the primal-dual schemahave 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 betterthan the integrality gap of the relaxation. Therefore integer programs with small integrality gap are criticalto the success of such approximation algorithms. The immediate program is i) to develop relaxations withbounded 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 strategiesfor converting the fractional solutions to the relaxations to integral solution. There has been considerableresearch 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 approximationalgorithms 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 aspart 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 qualityjournals and conferences. The program will produce highly skilled manpower; skilled in the use of discreteoptimization theory and tools.

一般认为，不存在有效的算法来寻找NP难优化问题的最优解。一个自然的方式来处理无法找到精确的解决方案，有效地是贸易的质量解决方案的计算时间。近似算法正是这样做的。近似算法不仅在多项式时间内提供近似解，而且还为解提供最优性证明。人们总是可以改进和调整一个近似算法，以适应实际中出现的特定类别的情况，从而提高性能比。我们将为生物信息学、设施定位、机器学习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在组合算法、线性和半定规划中具有理论基础。基于线性规划的方法在最近的过去取得了很大的成功。该方法的基本框架是将优化问题描述为整数变量上的整数线性规划或非线性规划，并构造了相应的线性规划松弛或半定规划松弛.使用有效的算法来解决该问题。这样得到的分数阶解用某种方法转化为积分解。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-对偶模式迭代地构造整数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，而不必求助于分离预言机。基于原始-对偶模式的方法对于某些类型的优化问题已经非常成功。整数规划公式的积分间隙是最优分式代价和最优积分代价的差距，基于线性或半定规划的逼近算法的性能比不优于松弛的积分间隙。因此，具有小整数间隙的整数规划是这类近似算法成功的关键.直接的程序是：i）为感兴趣的优化问题或表明不存在的优化问题开发具有有界积分间隙的松弛，ii）在可能的情况下开发求解松弛的组合算法，iii）开发可证明的将分数解转换为松弛解的良好策略。近几年来，对逼近的难性进行了大量的研究，得到了一些关于逼近的难性的深入结果。本研究的重点是近似算法的设计，我们将利用近似算法的困难性的结果来指导程序的设计，在理论上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的前景，并将对工业产生直接影响。所产生的知识将受到专利保护（如适用），并在高质量的期刊和会议上传播。该计划将产生高度熟练的人力;熟练使用离散优化理论和工具。