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CS-153 · computer science · mathematics · Algorithms and data structures

Worst-case Subset Sum below the half exponent

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $a_1,\ldots,a_n$ be positive integers, let $t\ge0$, and assume $n\ge2$. The Subset Sum decision asks whether $\sum_{i\in I}a_i=t$ for some $I\subseteq[n]$. Put \[ b=\max\{1,\lceil\log_2(1+\max(t,a_1,\ldots,a_n))\rceil\},\qquad w=\lceil4(n+b+\log_2(n+2))\rceil. \] Use a classical word RAM with $w$-bit words, read-only random-access input, charged word reads and writes, addressing, comparisons, arithmetic, bit operations and shifts. A fresh uniform random word costs one operation, and retained random words occupy writable memory. Seek a single uniform randomized program that halts on every execution, answers correctly with probability at least $2/3$ on every input, and satisfies \[ \forall c\in\mathbb N_{\ge1}\ \exists C_c>0,N_c\ \forall n\ge N_c,\ b\le n^c:\quad T\le C_c2^{0.49n} \] on every random path. All preprocessing is charged, and the program receives no value of $c$.

Short English statement

Can a uniform randomized Subset Sum solver run in $O_c(2^{0.49n})$ word operations for polynomial-bit inputs, using the specified $w$-bit RAM model and one program independent of $c$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-153-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-153

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 138; 2026-10-04).

Worst-case Subset Sum below the half exponent

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 138; 2026-10-04), the asserted result is that bounded-error classical Subset Sum on polynomial-bit inputs has a uniform $O(2^{0.49n})$ algorithm with that bound on every random execution.

Mathematical statement of the claimed result

Manuscript 1.
For positive integers $a_1,\ldots,a_n$, nonnegative target $t$ and $n\ge2$, ask whether $\sum_{i\in I}a_i=t$ for some subset. Put $b=\max\{1,\lceil\log_2(1+\max(t,a_1,\ldots,a_n))\rceil\}$ and use the stated word-RAM with read-only input and $w=\lceil4(n+b+\log_2(n+2))\rceil$-bit words; all preprocessing, addressing, word arithmetic and random generation are charged. The claimed single uniform randomized algorithm halts on every execution, is correct with probability at least $2/3$ on every input, and obeys \[ \forall c\ge1\ \exists C_c,N_c\ \forall n\ge N_c,\ b\le n^c: \quad T\le C_c2^{0.49n} \] word operations on every random path. The program is independent of the fixed polynomial bit-length exponent $c$.

Outline of the argument

For manuscript 1 (family 138; 2026-10-04), filtered representations recover modular aliases through small digit lists. Input-independent pair scores satisfy weighted Fourier-moment bounds, and checksum estimation combines direct sums with importance sampling; deterministic caps control both the visited lists and numerical work.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 138; 2026-10-04).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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