涉及拷贝数分布的基因组问题：复杂性与算法
Genomic Problems Involving Copy Number Profiles: Complexity and Algorithms
摘要：
近年来，由于多种癌症的基因组序列分析，基于{ \em拷贝数谱}（{ \em CNP}）的基因组数据越来越流行。CNP是一种载体，其中每个组分是代表特定基因或目的片段拷贝数的非负整数。在本文中，我们提出了两个结果流。首先是关于Qingge等人在2018年提出的最小拷贝数生成（MCNG）问题的计算复杂性的两个开放问题的负面结果。Qingge等人证明，如果重复是串联的，则问题是NP难的，并且他们留下了一个悬而未决的问题，即如果使用任意重复，则问题是否仍然是NP难的。我们肯定地回答这个问题，在本文中，事实上，我们证明，这是NP-困难的，甚至获得一个常数因子近似。我们还证明了参数化版本是W[1]-困难的，回答了Qingge等人的另一个公开问题。另一个结果是肯定的，并且基于一个新的（更一般的）关于CNP的问题。\emph{Copy Number Profile Conforming (CNPC)}问题的形式定义如下：给定两个CNP的$C_1$和$C_2$，用$cnp(S_1)=C_1$和$cnp(S_2)=C_2$计算两个字符串$S_1$和$S_2$，使得$S_1$和$S_2$之间的距离$d(S_1,S_2)$最小。在这里，$d(S_1,S_2)$是一个非常通用的术语，这意味着它可以是任何基因组重排距离（如逆转，转座和串联重复等）。我们的第一步表明，如果$d(S_1,S_2)$是由断点距离测量，那么这个问题是多项式可解的。
英文摘要：
Recently, due to the genomic sequence analysis in several types of cancer, the genomic data based on {\em copy number profiles} ({\em CNP} for short) are getting more and more popular. A CNP is a vector where each component is a non-negative integer representing the number of copies of a specific gene or segment of interest. In this paper, we present two streams of results. The first is the negative results on two open problems regarding the computational complexity of the Minimum Copy Number Generation (MCNG) problem posed by Qingge et al. in 2018. It was shown by Qingge et al. that the problem is NP-hard if the duplications are tandem and they left the open question of whether the problem remains NP-hard if arbitrary duplications are used. We answer this question affirmatively in this paper; in fact, we prove that it is NP-hard to even obtain a constant factor approximation. We also prove that the parameterized version is W[1]-hard, answering another open question by Qingge et al. The other result is positive and is based on a new (and more general) problem regarding CNP's. The \emph{Copy Number Profile Conforming (CNPC)} problem is formally defined as follows: given two CNP's $C_1$ and $C_2$, compute two strings $S_1$ and $S_2$ with $cnp(S_1)=C_1$ and $cnp(S_2)=C_2$ such that the distance between $S_1$ and $S_2$, $d(S_1,S_2)$, is minimized. Here, $d(S_1,S_2)$ is a very general term, which means it could be any genome rearrangement distance (like reversal, transposition, and tandem duplication, etc). We make the first step by showing that if $d(S_1,S_2)$ is measured by the breakpoint distance then the problem is polynomially solvable.
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