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CS-157-A1 · revision r1 · on CS-157

OpenAI claim summary for CS-157

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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 120; 2026-09-24).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Almost-linear maximum-cardinality matching
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Almost-linear maximum-cardinality matching

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 120; 2026-09-24), the asserted result is that general simple graphs admit a uniform randomized algorithm producing a maximum matching in $(n+m)^{1+o(1)}$ time on every path with success probability at least $2/3$, also yielding prescribed-degree subgraphs.

Mathematical statement of the claimed result

Manuscript 1.
For each finite simple general graph supplied as adjacency lists (a boundary array and two neighbor occurrences per undirected edge) $G=([n],E)$, put $m=|E|$, $p=n+m$ and let $\nu(G)$ be its maximum matching size. One uniform randomized word-RAM algorithm is claimed with absolute constants $C\ge1,c_0\ge2$ and a nonincreasing $\eta(p)\ge0$ tending to zero, using words of $\lceil c_0\log_2(n+2)\rceil$ bits, such that \[ T(G)\le Cp^{1+\eta(p)}\quad\hbox{on every random path},\qquad \Pr[\text{output is a matching of size }\nu(G)]\ge2/3. \] The matching is explicitly output as an edge list, and the graph need not be bipartite.

Outline of the argument

For manuscript 1 (family 120; 2026-09-24), factor-critical contraction and Hall reassignment maintain explicit matching states. Multiscale allocation ledgers control debt and resets; a cut-matching routing helper with conductance estimates keeps the required searches within the almost-linear budget.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 120; 2026-09-24).

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