Sorting a permutation by reversals is a famous problem in genome rearrangements. Since 1997, quite some biological evidence were found that in many genomes the reversed regions are usually flanked by a pair of inverted repeats. This type of reversals are called symmetric reversals, which, unfortunately, were largely ignored until recently. In this paper, we investigate the problem of sorting by symmetric reversals, which requires a series of symmetric reversals to transform one chromosome $A$ into the another chromosome $B$. The decision problem of sorting by symmetric reversals is referred to as {\em SSR} (when the input chromosomes $A$ and $B$ are given, we use {\em SSR(A,B)}) and the corresponding optimization version (i.e., when the answer for {\em SSR(A,B)} is yes, using the minimum number of symmetric reversals to convert $A$ to $B$), is referred to as {\em SMSR(A,B)}. The main results of this paper are summarized as follows, where the input is a pair of chromosomes $A$ and $B$ with $n$ repeats. (1) We present an $O(n^2)$ time algorithm to solve the decision problem {\em SSR(A,B)}, i.e., determine whether a chromosome $A$ can be transformed into $B$ by a series of symmetric reversals. (2) We design an $O(n^2)$ time algorithm for a special 2-balanced case of {\em SMSR(A,B)}, where chromosomes $A$ and $B$ both have duplication number 2 and every repeat appears twice in different orientations in $A$ and $B$. (3) We show that SMSR is NP-hard even if the duplication number of the input chromosomes are at most 2, hence showing that the above positive optimization result is the best possible. As a by-product, we show that the \emph{minimum Steiner tree} problem on \emph{circle graphs} is NP-hard, settling the complexity status of a 38-year old open problem.

翻译：通过反转对排列进行排序是基因组重排领域的一个著名问题。自1997年以来，大量生物学证据表明，在许多基因组中，反转区域通常两侧伴有一对反向重复序列。这种类型的反转被称为对称反转，遗憾的是，直到近期才引起广泛关注。本文研究了对对称反转排序问题，即通过一系列对称反转将一条染色体$A$转化为另一条染色体$B$。对称反转排序的决策问题称为{\em SSR}（当输入染色体$A$和$B$给定时，记为{\em SSR(A,B)}），相应的优化版本（即当{\em SSR(A,B)}的答案为“是”时，使用最少次数的对称反转将$A$转化为$B$）记为{\em SMSR(A,B)}。本文主要结果总结如下，其中输入为一对具有$n$个重复序列的染色体$A$和$B$：（1）我们提出了一种$O(n^2)$时间的算法来解决决策问题{\em SSR(A,B)}，即判断染色体$A$能否通过一系列对称反转转化为$B$。（2）针对{\em SMSR(A,B)}的一种特殊2-平衡情形，我们设计了一种$O(n^2)$时间的算法，在该情形下染色体$A$和$B$的重复数均为2，且每个重复在$A$和$B$中以不同方向出现两次。（3）我们证明，即使输入染色体的重复数不超过2，SMSR问题也是NP-难的，因此表明上述正面优化结果已是最优可能。作为副产品，我们证明了\emph{圆周图}上的\emph{最小斯坦纳树}问题是NP-难的，解决了一个存在38年之久的复杂性未决问题。