最大条带恢复的不可逼近性
Inapproximability of maximal strip recovery
摘要：
在比较基因组学中，序列分析的第一步通常是将两个或多个基因组分解成同源染色体的同线区段。为了可靠地恢复同线块，需要首先去除基因组图谱中的噪声和模糊性。最大条带恢复（Maximum Strip Recovery，MSR）是Zheng，Zhu和Sankoff提出的一个优化问题，用于在噪声和模糊性中从基因组图谱中可靠地恢复同线块。给定$d$基因组图谱作为基因标记的序列，\msr{d}的目标是找到$d$序列，每个基因组图谱的一个子序列，使得这些序列中同线块的总长度最大化。对于任何常数$d \ge 2$，以前已知\msr{d}的多项式时间2d近似。在本文中，我们表明，对于任何$d \ge 2$，\msr{d}是APX困难的，即使是最基本的版本的问题，其中所有的基因标记是不同的，并出现在每个基因组图谱的正方向。此外，我们提供了第一个明确的下界近似\msr{d}的所有$d \ge 2$。特别是，我们表明，\msr{d}是NP-难近似$Ω(d/\log d)$内。从另一个方向，我们表明，以前的二维近似\msr{d}可以优化成一个多项式时间算法，即使$d$是不是一个常数，但输入的一部分。然后，我们将我们的不可逼近性结果推广到几个相关的问题，包括\cmsr{d}，\gapmsr δ{d}，和\gapcmsr δ{d}。
英文摘要：
In comparative genomic, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given $d$ genomic maps as sequences of gene markers, the objective of \msr{d} is to find $d$ subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant $d \ge 2$, a polynomial-time 2d-approximation for \msr{d} was previously known. In this paper, we show that for any $d \ge 2$, \msr{d} is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating \msr{d} for all $d \ge 2$. In particular, we show that \msr{d} is NP-hard to approximate within $Ω(d/\log d)$. From the other direction, we show that the previous 2d-approximation for \msr{d} can be optimized into a polynomial-time algorithm even if $d$ is not a constant but is part of the input. We then extend our inapproximability results to several related problems including \cmsr{d}, \gapmsrδ{d}, and \gapcmsrδ{d}.
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