Result 128, Theoretical computer science

A factor-two approximation for shortest common superstring

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

Overlapping strings can be packed into one longer string that contains them all. The manuscript reports an efficient algorithm whose output is never more than twice as long as the best possible packing.

What changes?

A common superstring contains every input string as a contiguous block; the optimum is the shortest possible such string. For every finite family of explicitly represented ordinary strings, the manuscript claims a deterministic algorithm that constructs a common superstring of length at most twice the optimum. Its running time is polynomial in the full encoded input length, including symbol labels. This guarantee concerns the newly constructed algorithm, not the classical maximum-overlap Greedy procedure.

What does that help mathematicians do?

The claimed guarantee would let a researcher bound the best achievable packing without finding it exactly. If the algorithm returns a string of length L, the optimum must lie between half of L and L: the lower bound follows from the approximation guarantee, and the upper bound from the valid output itself. This gives a concrete interval for assessing how much further shortening could be possible.

Are there practical applications?

The relevant algorithmic connection is compact representation of overlapping strings while retaining every input as a contiguous segment. The demonstrated consequence claimed here is a worst-case bound on representation length, not a measured improvement on real datasets. Polynomial running time alone does not establish practical speed; the supplied abstract gives no implementation benchmarks or evidence of deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Polynomial-Time 2-Approximation for Shortest Common Superstring

September 24, 2026 19 pages Main result formalized in Lean

We give a deterministic algorithm that, for every finite family of explicitly represented ordinary strings, outputs a common superstring of length at most twice the optimum in time polynomial in the total encoded input size, including symbol labels. The guarantee applies to the algorithm constructed here, not the classical maximum-overlap Greedy procedure.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026,
  author = {{OpenAI}},
  title = {{A Polynomial-Time 2-Approximation for Shortest Common Superstring}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf}{OAI:A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/128.md.

A factor-two approximation for shortest common superstring

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The shortest common superstring problem asks for the shortest string containing each input string as a contiguous substring. The formalized result gives one deterministic polynomial-time algorithm whose output is a common superstring of length at most twice the unrestricted optimum. It applies to every finite list of explicitly encoded strings, with running time measured in input bits and the approximation ratio measured in symbol length.

Comparator links

Result Comparator statement
Polynomial-time 2-approximation for shortest common superstring Superstring.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.