Result 138, Theoretical computer science

Subset Sum in $O(2^{0.49n})$ time

Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary $O(2^{0.49n})$ word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have $O(n+b)$ bits for maximum input bit length b.

Algorithm or complexity result

The bigger picture

Why it matters

Subset Sum asks whether some of a given collection of integers add up exactly to a target. The manuscript claims a smaller exponential running-time bound, sharpening our understanding of the computational cost of this basic search problem.

What changes?

The manuscript reports a uniform randomized classical algorithm with worst-case time O(2^(0.49n)), where n counts the input integers. It allows repeated positive integers with polynomially bounded bit lengths. Computation uses a word-RAM model, which operates on words of O(n+b) bits, with b the largest input bit length. The time bound holds on every random execution, not just on average, and the probability of a correct answer is at least 2/3 for every input.

What does that help mathematicians do?

The claimed time dependence falls below 2 to the power n/2 even for worst-case instances, without restricting inputs to a favorable random distribution. This provides a stronger upper bound on the work needed to decide Subset Sum with bounded error. The separate low-space manuscript reports polynomial-in-n times 2^(n/2) time and O(2^(n/5)) writable words; its space guarantee should not be attributed to the faster algorithm.

Are there practical applications?

The immediate relevance is foundational: a sharper theoretical bound for exact combinatorial search, where selected items must meet a total exactly. The reported running time remains exponential and depends on the stated machine model. The abstracts provide no benchmarks, so they do not establish when this algorithm would outperform alternatives in practice.

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.

2 manuscripts

Subset Sum in Time $O(2^{0.49n})$

October 4, 2026 33 pages

We give a uniform randomized classical algorithm for Subset Sum with bounded error and worst-case running time $O(2^{0.49n})$ on polynomial-bit inputs in a word-RAM model, where n is the number of input integers. The time bound holds on every random execution.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Subset-Sum-in-Time-2-power-0-49n-October-4-2026,
  author = {{OpenAI}},
  title = {{Subset Sum in Time $O(2^{0.49n})$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf}{OAI:Subset-Sum-in-Time-2-power-0-49n-October-4-2026}},
  year = {2026}
}

A Low-Space Algorithm for Worst-Case Subset Sum

September 26, 2026 36 pages

We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses $\mathop{\mathrm{poly}}\nolimits (n)2^{n/2}$ time and ordinary $O(2^{n/5})$ writable words of $O(n+b)$ bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026,
  author = {{OpenAI}},
  title = {{A Low-Space Algorithm for Worst-Case Subset Sum}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf}{OAI:A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026}},
  year = {2026}
}

At a glance

Subject: Theoretical computer science
Manuscripts: 2
Pages: 69
Dated: Sep 26 to Oct 4, 2026
Lean formalization: No
Reasoning summary: No

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