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CS-129 · computer science · mathematics · Computational complexity · Graph theory and combinatorics

Constant-factor hardness of directed feedback vertex set

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed $A\geq 1$, is finding a directed feedback vertex set within factor A NP-hard on explicitly given unweighted digraphs, with a positive threshold and a deterministic polynomial-time gap reduction?

Short English statement

For every fixed $A\geq 1$, is finding a directed feedback vertex set within factor A NP-hard on explicitly given unweighted digraphs, with a positive threshold and a deterministic polynomial-time gap reduction?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-129-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-129

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 directed feedback vertex set

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 directed feedback vertex set is NP-hard to approximate within any prescribed constant, even without weights.

Mathematical statement of the claimed result

Manuscript 1.
Let $\operatorname{DFVS}(G)$ be the minimum number of vertices whose removal leaves a directed graph acyclic. For each fixed real $A\ge1$, the manuscript claims that an NP-hard promise problem can be mapped in deterministic polynomial time to an unweighted digraph $G$ and positive integer $k$, with the two resulting cases \[ \operatorname{DFVS}(G)\le k\qquad\hbox{from}\qquad\operatorname{DFVS}(G)>Ak. \] The polynomial may depend on $A$. This implies NP-hardness of every fixed-factor approximation for directed feedback vertex set.

Outline of the argument

For manuscript 1 (family 102; 2026-09-23), rank and wildcard vertices encode a collection of ordering tests. A low-cost deletion set leaves a topological order whose pivotal-label budget and shared-prefix conditioning decode an outer-game strategy; unweighted realization preserves the gap.

Formal verification

For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: directed feedback vertex set is NP-hard to approximate within any prescribed constant, even without weights.

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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