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Entropy and Face Dimension of the Perfect-Matching Polytope

OpenAI

For a loopless multigraph on 2m vertices with a perfect matching and feasible edge marginals x, the maximum entropy H(x) satisfies F(x)−(2−2/m)B(x)≤H(x)≤F(x), where F and B sum the binary-coordinate entropy terms. Optimizing gives the linear-error estimate proposed by Anari, Oveis Gharan and Vinzant.

Sources for this record

What was already known

The proposed convex-optimization estimate sought a linear-in-vertices error for general-graph perfect matchings. Stronger weighted Bethe comparisons were known for bipartite graphs.

AI involvement AI-led

OpenAI attributes the central inequality to an internal model; autonomy of this individual research run is undocumented.

The central entropy inequality is in the model-produced manuscript; autonomy of this individual run is undocumented.

Importance Notable

Claims the proposed linear-error estimate for general-graph perfect matchings.

Limitations

The exact pointwise coefficient remains unverified in the released Lean comparator; a completed Lean file proves a weaker nonlinear defect bound. The deterministic counting algorithm in the manuscript is a separate result.

Technical verification Reported

The manuscript states and proves the sharp pointwise bound, but the released comparator for that statement still has sorry. A separate completed Lean file proves a weaker nonlinear defect and counting bound, not this precise inequality. No complete human check of the assessed claim is documented, so Reported is the evidence default.

Human understanding Few

No named human explanation of the tight odd-cut face-dimension and entropy-duality mechanism is documented; Few is the evidence default.

Correction and publication history