串联重复距离问题在有界字母表上是困难的
The Tandem Duplication Distance Problem is hard over bounded alphabets
摘要：
串联复制指的是将DNA片段的拷贝插入其原始位置附近的过程。更正式地说，串联重复可以被认为是将字符串$S = AXB$转换为字符串$T = AXXB.$的操作。此外，串联复制机制最近已经在不同的背景下，从形式语言，信息论，DNA存储系统的纠错码的研究。确定计算两个给定字符串之间的串联重复距离的复杂性的问题由[Leupold等人，2004年]，最近，它被证明是NP-困难的情况下，无限的字母[Lafond等人，STACS2020]。在本文中，我们显着改善这一结果，并表明，串联重复距离问题是NP-困难的情况下，字符串的大小为$\leq 5.$我们还研究了一些特殊类别的字符串是可能的线性时间解决方案的存在性问题：给定相同字母表上的字符串$S$和$T$，判定是否存在将$S$转换为$T$的重复序列。一个解决存在性问题的多项式时间算法只在二进制字母表的情况下才知道。
英文摘要：
A tandem duplication denotes the process of inserting a copy of a segment of DNA adjacent to its original position. More formally, a tandem duplication can be thought of as an operation that converts a string $S = AXB$ into a string $T = AXXB.$ As they appear to be involved in genetic disorders, tandem duplications are widely studied in computational biology. Also, tandem duplication mechanisms have been recently studied in different contexts, from formal languages, to information theory, to error-correcting codes for DNA storage systems. The problem of determining the complexity of computing the tandem duplication distance between two given strings was proposed by [Leupold et al., 2004] and, very recently, it was shown to be NP-hard for the case of unbounded alphabets [Lafond et al., STACS2020]. In this paper, we significantly improve this result and show that the tandem duplication distance problem is NP-hard already for the case of strings over an alphabet of size $\leq 5.$ We also study some special classes of strings were it is possible to give linear time solutions to the existence problem: given strings $S$ and $T$ over the same alphabet, decide whether there exists a sequence of duplications converting $S$ into $T$. A polynomial time algorithm that solves the existence problem was only known for the case of the binary alphabet.
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