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

拟议的研究计划的重点是设计高效的算法与证明性能的计算优化问题的启发，真实的世界的应用程序，特别是从多组学研究和运筹学。当面对一个具有挑战性的真实的世界的应用程序，典型的过程是首先制定的问题数学，追求一个指定的目标函数的优化。然后研究优化问题，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，我们的大多数目标问题所产生的具有挑战性的应用程序不能在多项式时间内最佳地解决，除非P = NP。我们遵循主流的研究路线，研究问题的不可逼近性和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，并寻求开拓固定参数可逼近性。近似算法在多项式时间内运行，并产生保证在最优解的某个因子内的解。近似算法的设计和分析研究具有多方面的影响：1）由于目标优化问题的复杂性，近似算法至少提供了一种找到具有可证明保证的近似最优解的方法：2）尽管最坏情况下的性能保证可能令人失望，但近似算法在真实的世界实例上往往表现得很好;以及3）最重要的是，所开发的算法工具和设计与分析技术通常是有用的，即使近似算法本身可能不是非常实用。虽然近似算法是逼近NP难问题的积极结果，但研究这种近似的极限也很重要，称为近似的硬度或不可近似性。通过证明问题的可逼近性的下界，我们对NP难优化问题的频谱有了更深的了解。这些见解可以反过来用于开发新的和更好的近似算法。问题的可处理性和可近似性可以随着（通常是多个）参数和/或输出而沿着改变。固定参数的可处理性已被广泛研究，但结果的逼近问题的变化沿着在输入或输出中的参数是有限的。对运行时间和性能比均取决于参数k的近似算法的研究具有与上述相同的多方面方面的方面，并且还提供了对参数k的见解以及随后对目标优化问题的内部谱的另一种更深入的理解。

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.