Exact ratios for fixed-length shortest common superstring instances

Determine the exact worst-case approximation ratio \(\rho_k\) of the greedy superstring algorithm for each fixed string length \(k\ge 3\) not completely characterized by the paper.

Background

For the restriction in which every input string has length exactly kk, the paper denotes the greedy algorithm's worst-case approximation ratio by ρk\rho_k. Before this work, only bounds were known for the fixed-length cases. The paper determines ρ3=9/5\rho_3=9/5 and proves ρk≥2\rho_k\ge 2 for all k≥6k\ge 6, but it does not determine the exact ratios for the remaining unresolved cases or settle the general fixed-length behavior.

The unresolved fixed-length question is connected to the broader greedy superstring conjecture: knowing the exact values of ρk\rho_k, or sufficiently sharp bounds for all kk, would clarify the asymptotic behavior of the greedy algorithm on bounded-length inputs.

References

The conjecture is~open for~$40$~years already and even the approximation ratio~$\rho_k$ in~the special case when input strings have length~$k$ has~not yet been found: for all $k \ge 3$, $2-1/k \le \rho_k \le \min{(k+1)/2, 3.396}$.

— The Greedy Superstring Algorithm Achieves Ratio 2 for Strings of Length 6 Already  (2608.20018 - Chukhin et al., 20 Aug 2026) in Abstract and Section 1, "Greedy Superstring Conjecture"