Improved Approximation Ratios for the Shortest Common Superstring Problem with Reverse Complements

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.

• The paper presents improved approximation ratios for the Shortest Common Superstring Problem with Reverse Complements (SCS-RC) algorithms MGREEDY-RC and TGREEDY-RC.
• Specifically, MGREEDY-RC's approximation ratio is improved from 4 to 3.75, and TGREEDY-RC's ratio from 3 to 2.875, marking the first formal improvements since Jiang et al.'s analysis.
• The work leverages a new framework that utilizes pairwise overlaps between orientation pairs and then averages the bounds over an optimal solution and its reverse complement.
• follow_up_questions

Background and problem statement

The Shortest Common Superstring (SCS) problem asks for a shortest string containing every string of an input set $S$ as a substring. Its reverse-complement variant, SCS-RC, requires that for each $s_i \in S$, either $s_i$ or its reverse complement $\bar{s_i}^R$ appear in the superstring. This variant models DNA sequence assembly, where reads are double-stranded and their orientation is unknown. Despite three decades of progress on standard SCS — from Blum et al.'s 4-approximation through successive improvements culminating in Englert et al.'s $\frac{\sqrt{67}+14}{9} \approx 2.465$ bound — the reverse-complement setting had remained at the guarantees established by Jiang et al.: 4-approximation for MGREEDY-RC and 3-approximation for TGREEDY-RC.

The algorithms

The paper analyzes three greedy algorithms adapted to the reverse-complement setting:

• MGREEDY-RC: repeatedly merges pairs $(x, y)$ from $S \cup \bar{S}^R$ with maximum overlap, explicitly forbidding merges of the form $(x, \bar{x}^R)$ when $x \neq \bar{x}^R$, since both orientations need not co-occur; strings that cannot be merged with anything (self-overlap case) are moved to a set $T$. The algorithm implicitly constructs an optimal cycle cover $s_i \in S$0 of the distance graph over $s_i \in S$1, where each cycle contains exactly one orientation of each input string.
• GREEDY-RC: repeatedly merges maximum-overlap pairs until one string remains; Fici et al. proved it achieves compression ratio $s_i \in S$2.
• TGREEDY-RC: runs MGREEDY-RC to obtain $s_i \in S$3, then applies GREEDY-RC to merge the resulting cycle strings rather than concatenating them directly.

The analysis works with the distance graph $s_i \in S$4 weighted by $s_i \in S$5, and uses the fact that any optimal SCS-RC solution has the form $s_i \in S$6 where each $s_i \in S$7 is either $s_i \in S$8 or $s_i \in S$9. A key structural fact carried over from the standard setting is that distinct cycles in an optimal cycle cover yield pairwise inequivalent periodic strings — including under reverse complementation.

The general framework theorem

The first contribution is a reduction framework analogous to Theorem 3.1 of Englert et al. for standard SCS: if MGREEDY-RC is a $s_i$0-approximation and some algorithm achieves compression ratio $s_i$1 for SCS-RC, then one can construct a $s_i$2-approximation. The proof shows that $s_i$3 by expanding representative strings into full cycle strings within an optimal superstring, exploiting that each representative or its reverse complement occurs in the optimum so that replacement remains well-defined. Since GREEDY-RC gives $s_i$4, TGREEDY-RC inherits a $s_i$5 guarantee. Consequently, every improvement to MGREEDY-RC's ratio translates at half strength to TGREEDY-RC.

Improving MGREEDY-RC to 3.75

The main technical obstacle is that Kaplan and Shafrir's improvement of MGREEDY from 4 to 3.5 relies on the overlap rotation lemma of Breslauer et al., which bounds overlaps involving a critical rotation of a periodic semi-infinite string by $s_i$6. In the reverse-complement setting this lemma fails directly: if the rotated occurrence appears as $s_i$7 rather than $s_i$8, only the weaker inequivalence bound $s_i$9 applies.

The paper's resolution proceeds in three steps. First, for each cycle $\bar{s_i}^R$0 in the optimal cycle cover, extract the critical-rotation string $\bar{s_i}^R$1 guaranteed by Breslauer et al., and show via the greedy maximality of the cycle-closing edge that $\bar{s_i}^R$2 where $\bar{s_i}^R$3. Second, bound $\bar{s_i}^R$4 using the containing strings $\bar{s_i}^R$5. Third, bound the total inter-string overlap $\bar{s_i}^R$6 along an optimal ordering of $\bar{s_i}^R$7: occurrences of $\bar{s_i}^R$8 contribute $\bar{s_i}^R$9 via the rotation lemma, while occurrences of $\frac{\sqrt{67}+14}{9} \approx 2.465$0 contribute $\frac{\sqrt{67}+14}{9} \approx 2.465$1 via the weak lemma. The crucial observation is that both an optimal superstring $\frac{\sqrt{67}+14}{9} \approx 2.465$2 and its reverse complement $\frac{\sqrt{67}+14}{9} \approx 2.465$3 are optimal for the same instance, and they swap the roles of $\frac{\sqrt{67}+14}{9} \approx 2.465$4 and $\frac{\sqrt{67}+14}{9} \approx 2.465$5; averaging the two inequalities yields

$\frac{\sqrt{67}+14}{9} \approx 2.465$6

hence $\frac{\sqrt{67}+14}{9} \approx 2.465$7.

Main results. Combining these pieces, the paper proves:

Algorithm	Previous ratio	New ratio
MGREEDY-RC	4	3.75
TGREEDY-RC	3	2.875

The 2.875 bound for TGREEDY-RC is the best known approximation guarantee for SCS-RC, and these are the first formal improvements since Jiang et al.'s original analysis. Note that the averaging argument caps the benefit of the rotation lemma at half its full effect: in the standard setting Kaplan and Shafrir obtained 3.5, whereas here the unavoidable weak-lemma contributions limit MGREEDY-RC to 3.75.

Limitations and open questions

The paper identifies two specific gaps relative to standard SCS. First, the best known ratio for MGREEDY on standard SCS is $\frac{\sqrt{67}+14}{9} \approx 2.465$8 (Englert et al.), versus 3.75 here; whether the more refined cycle-classification techniques of Englert et al. can be ported to the reverse-complement setting remains open. Second, the framework's second component uses GREEDY-RC's $\frac{\sqrt{67}+14}{9} \approx 2.465$9 compression ratio, whereas standard SCS admits a $(x, y)$0-compression algorithm via reduction to MAX-ATSP. For SCS-RC, vertices come in orientation pairs, forming clusters of two vertices, which blocks a direct MAX-ATSP reduction; finding an appropriate reduction or an alternative route to a better compression ratio is left open. The paper also does not establish matching lower bounds for either algorithm, so the true worst-case ratios of MGREEDY-RC and TGREEDY-RC remain undetermined.

Conclusion

This work closes part of a long-standing gap between standard SCS and its reverse-complement variant by proving that MGREEDY-RC is 3.75-approximate and TGREEDY-RC is 2.875-approximate. Technically, it adapts the critical-rotation machinery to bidirectional overlaps by averaging overlap bounds over an optimal solution and its reverse complement, thereby halving the penalty incurred where the rotation lemma does not apply. The result establishes the current state of the art for SCS-RC and delineates precisely which components of the modern standard-SCS analysis — tighter MGREEDY ratios and MAX-ATSP-based compression — do not yet transfer to the reverse-complement setting.