The Shortest Common Superstring (SCS) problem is a fundamental task in sequence analysis. In genome assembly, however, the double-stranded nature of DNA implies that each fragment may occur either in its original orientation or as its reverse complement. This motivates the Shortest Common Superstring with Reverse Complements (SCS-RC) problem, which asks for a shortest string that contains, for each input string, either the string itself or its reverse complement as a substring. The previously best-known approximation ratio for SCS-RC was $\frac{23}{8}$. In this paper, we present a new approximation algorithm achieving an improved ratio of $\frac{8}{3}$. Our approach computes an optimal constrained cycle cover by reducing the problem, via a novel gadget construction, to a maximum-weight perfect matching in a general graph. We also investigate the computational hardness of SCS-RC. While the decision version is known to be NP-complete, no explicit inapproximability results were previously established. We show that the hardness of SCS carries over to SCS-RC through a polynomial-time reduction, implying that it is NP-hard to approximate SCS-RC within a factor better than $\frac{333}{332}$. Notably, this hardness result holds even for the DNA alphabet.

翻译：最短公共超串（SCS）问题是序列分析中的一项基础性任务。然而，在基因组组装中，DNA的双链特性意味着每个片段可能以其原始方向或其反向互补形式出现。这催生了带反向互补的最短公共超串（SCS-RC）问题，该问题要求找到一条最短的字符串，使得对于每个输入字符串，其本身或其反向互补作为子串出现。此前SCS-RC已知的最佳近似比为$\frac{23}{8}$。本文提出一种新的近似算法，将近似比改进至$\frac{8}{3}$。我们的方法通过一种新颖的构件构造，将问题归结为一般图中的最大权重完美匹配，从而计算出最优约束环覆盖。我们还研究了SCS-RC的计算难度。虽然其判定版本已知是NP完全的，但此前未建立明确不可近似性结果。我们通过多项式时间归约表明，SCS的难度可传递至SCS-RC，这意味着在因子优于$\frac{333}{332}$的范围内近似SCS-RC是NP难的。值得注意的是，该难度结果甚至对DNA字母表也成立。