A Tight Cycle-Cover Inequality for Shortest Common Superstring

2026-08-21

AI Summary

This study addresses the long-standing challenge that upper bounds on the approximation ratio for the Shortest Common Superstring (SCS) problem have remained suboptimal, while tight theoretical performance guarantees for the greedy algorithm are lacking. By leveraging overlap graph modeling and combinatorial optimization analysis, this work pushes key cycle cover inequalities to their theoretical limit for the first time. It rigorously proves the unimprovability of specific coefficient upper bounds, thereby establishing new tight theoretical limits. Consequently, this research overcomes existing theoretical bottlenecks by improving the general approximation ratio for SCS to 7/3 and reducing the approximation ratio of the greedy algorithm from 3.396 to 3, setting new state-of-the-art records in the field.

Abstract

In the Shortest Common Superstring problem (SCS), one is given a finite set of strings and is asked to find a shortest string containing every input string as a substring. Its best known approximation ratio is $2.466$, whereas the currently strongest upper bound on the approximation guarantee of the maximum-overlap greedy algorithm is $3.396$ (Englert, Matsakis, and Vesel{\'y}, 2023), though it is conjectured to be $2$. We improve both approximation guarantees: SCS admits a $\frac{7}{3}$ approximation and the approximation guarantee of the greedy algorithm is at most $3$. The main technical ingredient of our proof is a certain inequality for minimum-cost cycle covers of an overlap graph associated with the input strings. Every previous improvement of greedy's worst-case guarantee and the two recent record guarantees for general SCS are driven by it. We improve this inequality by pushing it to its limit: for a particular coefficient of this inequality, we show a new upper bound and prove that it cannot be improved further.

Problem

• Shortest Common Superstring
• Approximation Ratio
• Greedy Algorithm
• Cycle Cover

Innovation

• Shortest Common Superstring
• Cycle-Cover Inequality
• Approximation Ratio
• Greedy Algorithm
• Overlap Graph