DNA片段组装的异常值检测
Outlier Detection for DNA Fragment Assembly
摘要：
给定$n$长度-恒定大小字母表$Σ$上的$\ell$字符串$S =\{s_1, ..., s_n\}$以及参数$d$和$k$，{具有离群值的\em共识字符串}问题中的目标是找到大小为$n-k$的$S$的子集$S^*$和字符串$s$，使得$\sum_{s_i \in S^*} d(s_i, s) \leq d$。这里$d(x, y)$表示两个字符串$x$和$y$之间的汉明距离。我们证明1. {具有异常值的\em共识串}的一个变体，其中异常值$k$的数量是固定的，并且目标是最小化总距离$\sum_{s_i \in S^*} d(s_i, s)$，允许简单的PTAS。(ii)在离群值$k$的数量很小的自然假设下，{具有离群值的\em共识字符串}的距离最小化版本的PTAS表现良好。特别地，只要$k\leq cn$为固定常数$c < 1$，该算法就提供了时间上的$(1+ε)$近似解$f(1/ε)(n\ell)^{O(1)}$，因此是EPTAS。2.为了将{具有异常值的\em共识字符串}的PTAS改进为EPTAS，需要假设$k$很小。具体来说，当$k$被允许是任意的时，{具有异常值的\em共识字符串}问题不允许EPTAS，除非FPT=W[1]。这个硬性结果甚至对二进制字母表也成立。3.当通过$\frac{d}{n-k}$参数化时，{具有离群值的\em共识字符串}的决策版本是固定参数可处理的。因此，当仅由$d$参数化时也是如此。据我们所知，{ \em共识字符串与离群值}是第一个问题，承认一个PTAS，是固定参数时，参数化的目标函数的值，但不承认合理的复杂性假设下的EPTAS。
英文摘要：
Given $n$ length-$\ell$ strings $S =\{s_1, ..., s_n\}$ over a constant size alphabet $Σ$ together with parameters $d$ and $k$, the objective in the {\em Consensus String with Outliers} problem is to find a subset $S^*$ of $S$ of size $n-k$ and a string $s$ such that $\sum_{s_i \in S^*} d(s_i, s) \leq d$. Here $d(x, y)$ denotes the Hamming distance between the two strings $x$ and $y$. We prove 1. a variant of {\em Consensus String with Outliers} where the number of outliers $k$ is fixed and the objective is to minimize the total distance $\sum_{s_i \in S^*} d(s_i, s)$ admits a simple PTAS. (ii) Under the natural assumption that the number of outliers $k$ is small, the PTAS for the distance minimization version of {\em Consensus String with Outliers} performs well. In particular, as long as $k\leq cn$ for a fixed constant $c < 1$, the algorithm provides a $(1+ε)$-approximate solution in time $f(1/ε)(n\ell)^{O(1)}$ and thus, is an EPTAS. 2. In order to improve the PTAS for {\em Consensus String with Outliers} to an EPTAS, the assumption that $k$ is small is necessary. Specifically, when $k$ is allowed to be arbitrary the {\em Consensus String with Outliers} problem does not admit an EPTAS unless FPT=W[1]. This hardness result holds even for binary alphabets. 3. The decision version of {\em Consensus String with Outliers} is fixed parameter tractable when parameterized by $\frac{d}{n-k}$. and thus, also when parameterized by just $d$. To the best of our knowledge, {\em Consensus String with Outliers} is the first problem that admits a PTAS, and is fixed parameter tractable when parameterized by the value of the objective function but does not admit an EPTAS under plausible complexity assumptions.
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