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CS-144 · computer science · mathematics · Computational complexity · Graph theory and combinatorics

Constant-factor hardness of Min-UnCut

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed integer $K\geq 2$, can 3SAT be reduced deterministically in polynomial time to an explicit finite simple unweighted graph and positive integer threshold k so that satisfiable inputs have minimum uncut-edge count $\leq k$ and unsatisfiable inputs have count $>Kk$?

Short English statement

For every fixed integer $K\geq 2$, can 3SAT be reduced deterministically in polynomial time to an explicit finite simple unweighted graph and positive integer threshold k so that satisfiable inputs have minimum uncut-edge count $\leq k$ and unsatisfiable inputs have count $>Kk$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-144-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-144

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 102; 2026-09-23).

Constant-factor hardness of Min-UnCut

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 102; 2026-09-23), the asserted result is that Min-UnCut admits no fixed constant-factor approximation unless the stated NP-hard distinction can be decided.

Mathematical statement of the claimed result

Manuscript 1.
Define $\operatorname{OPT}_{\rm uncut}(G)=\min_{S\subseteq V}|E(S)|+|E(V\setminus S)|$ for a finite simple undirected graph. For every fixed integer $K\ge2$, the claimed reduction transforms $3$SAT instances in deterministic polynomial time into $(G,k)$, with $G$ unweighted and binary integer $k\ge1$, such that \[ \varphi\text{ satisfiable}\Rightarrow\operatorname{OPT}_{\rm uncut}(G)\le k, \qquad\varphi\text{ unsatisfiable}\Rightarrow\operatorname{OPT}_{\rm uncut}(G)>Kk. \] For fixed $K$, all construction and bit costs are polynomial in formula length, without advice or an oracle.

Outline of the argument

For manuscript 1 (family 102; 2026-09-23), folded bit comparisons turn face correlation into a large odd Fourier coefficient. Alphabet-independent subbox decoding converts that coefficient into outer-game lists, and rational Gaussian tables are realized by an exact simple unweighted graph with a strict gap.

Formal verification

For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: Min-UnCut admits no fixed constant-factor approximation unless the stated NP-hard distinction can be decided.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are DirectedFeedback.lean, MinUncut.lean, OptimalMaxCut.lean, UniqueGamesTheorem.lean, VertexCover.lean.

Their named declarations include OAI.DirectedFeedback.main, OAI.MinUncut.main, OAI.OptimalMaxCut.main, OAI.UniqueGamesTheorem.theorem11, OAI.VertexCover.every_fixed_factor_below_two, OAI.VertexCover.explicit_cover_gap.

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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