Genome rearrangements are mutations affecting large portions of a genome, and a reversal is one of the most studied genome rearrangements in the literature through the Sorting by Reversals (SbR) problem. SbR is solvable in polynomial time on signed permutations (i.e., the gene orientation is known), and it is NP-hard on unsigned permutations. This problem (and many others considering genome rearrangements) models genome as a list of its genes in the order they appear, ignoring all other information present in the genome. Recent works claimed that the incorporation of the size of intergenic regions, i.e., sequences of nucleotides between genes, may result in better estimators for the real distance between genomes. Here we introduce the Sorting Signed Permutations by Intergenic Reversals problem, that sorts a signed permutation using reversals both on gene order and intergenic sizes. We show that this problem is NP-hard by a reduction from the 3-partition problem. Then, we propose a 2-approximation algorithm for it. Finally, we also incorporate intergenic indels (i.e., insertions or deletions of intergenic regions) to overcome a limitation of sorting by conservative events (such as reversals) and propose two approximation algorithms.

中文翻译：

---

通过基因间反转对有符号排列进行排序


基因组重排是影响基因组大部分的突变，而逆转是文献中通过逆转排序 (SbR) 问题研究最多的基因组重排之一。 SbR 在有符号排列上可以在多项式时间内求解（即基因方向已知），并且在无符号排列上是 NP 困难的。这个问题（以及许多其他考虑基因组重排的问题）将基因组建模为按其出现顺序排列的基因列表，忽略了基因组中存在的所有其他信息。最近的工作声称，基因间区域（即基因之间的核苷酸序列）大小的合并可能会产生更好的基因组之间真实距离的估计器。在这里，我们介绍了通过基因间反转对有符号排列进行排序问题，该问题使用基因顺序和基因间大小的反转来对有符号排列进行排序。我们通过 3 分区问题的简化证明这个问题是 NP 困难的。然后，我们提出了一个2-近似算法。最后，我们还结合了基因间插入缺失（即基因间区域的插入或删除）来克服保守事件（例如逆转）排序的限制，并提出了两种近似算法。