Approximation algorithms for hard optimization problems in multi-omics research and operations research

所提出的研究计划致力于为受现实世界应用，特别是来自多组学研究和运筹学研究的计算优化问题的启发，设计具有可证明性能的高效算法。当面对具有挑战性的现实应用时，典型的过程是首先对问题进行数学描述，追求指定目标函数的最优化。然后对优化问题进行研究，以了解其困难程度，并通过对其效率、性能和正确性的严格分析来设计算法。但是，除非P=NP，否则我们的大多数目标问题都不能在多项式时间内得到最优解决。我们沿着主流研究的路线来研究问题的非可逼近性，并设计问题的逼近算法。此外，我们将研究问题的固定参数可处理性，并试图开创固定参数可逼近的先河。*近似算法运行在多项式时间内，并产生保证在最优解的某一因子内的解。对近似算法的设计和分析的研究具有多方面的影响：1)由于目标优化问题的难解性，近似算法至少给出了一种找到具有可证明保证的近最优解的方法：2)尽管最坏情况的性能保证可能看起来令人失望，但近似算法在现实世界的实例中往往表现得很好；3)最重要的是，开发的算法工具和设计分析技术通常是有用的，即使近似算法本身可能不是很实用。*虽然近似算法对于逼近NP-Hard问题是积极的结果，但研究这种逼近的极限也很重要，即逼近的难度或不可逼近。通过证明问题的逼近下界，我们对NP-Hard优化问题的谱有了更深入的了解。这样的洞察力又可以用来开发新的和更好的近似算法。*问题的可处理性和可逼近性可以随着(通常是多个)参数和/或输出而改变。固定参数可处理性已被广泛研究，但关于问题的可逼近性如何随输入或输出中的参数变化的结果有限。对运行时间和性能比都依赖于参数k的近似算法的研究具有与上述相同的多方面的方面，并另外提供了对参数k的洞察以及随后对目标优化问题的内谱的另一更深层次的理解。

The proposed research program focuses on designing efficient algorithms with provable performance for computational optimization problems inspired by real world applications, in particular from multi-omics research and operations research. When facing a challenging real world application, the typical process is first to formulate the problem mathematically, to pursue the optimization of a specified objective function. The optimization problem is then studied to understand its level of difficulty and to design algorithms supported by a rigorous analysis of its efficiency, performance and correctness. But most of our target problems arising from challenging applications cannot be solved optimally in polynomial time, unless P = NP. We follow the line of main stream research to study the in-/approximability of the problems and to design approximation algorithms for the problems. Additionally, we will study the fixed parameter tractability of the problems and seek to pioneer the fixed parameter approximability.******Approximation algorithms run in polynomial time and produce solutions that are guaranteed to be within a certain factor of the optimal solution. The study of the design and analysis of approximation algorithms has multi-faceted impact: 1) due to the intractability of the target optimization problem, an approximation algorithm at least gives a way to find a near-optimal solution with provable guarantee; 2) although the worst-case performance guarantee may appear disappointing, an approximation algorithm can frequently perform really well on real world instances; and 3) most importantly, the developed algorithmic tools and design-and-analysis techniques can be generally useful, even if the approximation algorithm by itself may not be very practical.******While approximation algorithms are positive results for approaching an NP-hard problem, it is also important to study the limit of such approximation, known as the hardness of approximation or inapproximability. By proving lower bounds on approximability of the problems, we achieve deeper insights on the spectrum of the NP-hard optimization problems. Such insights can in turn be used to develop new and better approximation algorithms.******The tractability and the approximability of the problem could change along with the (often multiple) parameters and/or output. Fixed parameter tractability has been extensively investigated, but results on how the approximability of the problems changes along with the parameters in the input or output are limited. The study on approximation algorithms with both running time and performance ratio depending on a parameter k has the same multi-faceted aspects as stated in the above, and additionally provides insights on the parameter k and subsequently another deeper understanding of the internal spectrum of the target optimization problem.**