Improved Approximation Ratios for the Shortest Common Superstring Problem with Reverse Complements
Ryosuke Yamano, Tetsuo Shibuya
TL;DR
This work studies the Shortest Common Superstring problem with Reverse Complements (SCS-RC), a bioinformatics-relevant variant requiring for each input string either the string or its reverse complement to appear in the final superstring. It extends classical greedy approaches by adapting overlap-based analyses to bidirectional overlaps, culminating in new theoretical guarantees: MGREEDY-RC achieves a $($$3.75$$)$-approximation and TGREEDY-RC achieves a $($$2.875$$)$-approximation, the best known for SCS-RC to date. The authors introduce a framework that parallels the standard SCS framework, showing that a $(2+ abla)$-approximation MGREEDY-RC combined with a $\delta$-compression for SCS-RC yields a $(2+(1-\delta) abla)$-approximation for TGREEDY-RC, and apply it with $\delta=\tfrac{1}{2}$ to obtain the improved bounds. The results close part of the gap between SCS and SCS-RC guarantees and provide techniques—notably the overlap-rotation lemma—for handling reverse-complement overlaps in cycle-cover based analyses.
Abstract
The Shortest Common Superstring (SCS) problem asks for the shortest string that contains each of a given set of strings as a substring. Its reverse-complement variant, the Shortest Common Superstring problem with Reverse Complements (SCS-RC), naturally arises in bioinformatics applications, where for each input string, either the string itself or its reverse complement must appear as a substring of the superstring. The well-known MGREEDY algorithm for the standard SCS constructs a superstring by first computing an optimal cycle cover on the overlap graph and then concatenating the strings corresponding to the cycles, while its refined variant, TGREEDY, further improves the approximation ratio. Although the original 4- and 3-approximation bounds of these algorithms have been successively improved for the standard SCS, no such progress has been made for the reverse-complement setting. A previous study extended MGREEDY to SCS-RC with a 4-approximation guarantee and briefly suggested that extending TGREEDY to the reverse-complement setting could achieve a 3-approximation. In this work, we strengthen these results by proving that the extensions of MGREEDY and TGREEDY to the reverse-complement setting achieve 3.75- and 2.875-approximation ratios, respectively. Our analysis extends the classical proofs for the standard SCS to handle the bidirectional overlaps introduced by reverse complements. These results provide the first formal improvement of approximation guarantees for SCS-RC, with the 2.875-approximate algorithm currently representing the best known bound for this problem.