关于矩阵中值问题
On the Matrix Median Problem
摘要：
基因组中值问题是重排模型下系统发育重建中的一个重要问题。它可以表述如下：给定三个基因组，找到第四个使它与三个输入基因组之间的成对重排距离之和最小化的基因组。最近，Feijao和Meidanis扩展了基因组重排的代数理论，允许线性染色体，从而产生了一个新的重排模型（代数模型），非常接近著名的DCJ模型。本文研究了代数模型下的基因组中位数问题，提出了一种更一般的问题形式--矩阵中位数问题。已知对于任何度量距离，至少一个角是中值的4/3近似。我们的研究结果使我们能够计算多达三个额外的矩阵中位数候选人，所有这些都与近似比至少一样好的最好的角落，当输入矩阵来自基因组。从应用程序的角度来看，通常更有趣的是将中位数定位在离拐角较远的地方。我们还展示了第四个中位数候选人，在我们尝试的情况下，它给出了更好的结果。然而，我们还没有证明这第四个候选人的界限。
英文摘要：
The Genome Median Problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. Recently, Feijao and Meidanis extended the algebraic theory for genome rearrangement to allow for linear chromosomes, thus yielding a new rearrangement model (the algebraic model), very close to the celebrated DCJ model. In this paper, we study the genome median problem under the algebraic model, whose complexity is currently open, proposing a more general form of the problem, the matrix median problem. It is known that, for any metric distance, at least one of the corners is a 4/3-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. From the application point of view, it is usually more interesting to locate medians farther from the corners. We also show a fourth median candidate that gives better results in cases we tried. However, we do not have proven bounds for this fourth candidate yet.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
