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Sampling Directed Eulerian Tours in Õ(m³ᐟ²) Time

Nima Anari

Anari’s Theorem 4 reports that the flip–repair walk associated with any real skew-symmetric n×n matrix has spectral gap at least 2/n. In the flat 0/1 determinant case it also gives an entropy bound and nearly linear mixing; this feeds a near-uniform sampler for strongly connected directed Eulerian multigraphs with m arcs in O(m^(3/2) polylog(m/ε)) time.

Sources for this record

What was already known

Exact tour samplers relied on directed-arborescence sampling, with an mn-type runtime barrier on sparse graphs. Anari had already devised the direct walk, reduction and data structure and conjectured the needed mixing bounds.

AI involvement AI-centered

Anari credits GPT 5.5 Pro Extended with the square-root projection, excitation-number comparison and flat log-Sobolev proof. He set the target and devised the sampler architecture; Codex assembled and typeset the manuscript.

The model supplied the central missing mixing proof, while Anari formulated the claim and supplied the algorithmic framework and implementation.

Importance Notable

The mixing theorem supplies the missing step for a faster near-uniform Eulerian-tour sampler.

Limitations

No completed check of the model-generated mixing proof is documented. The full sampler bound also depends on Anari’s separate reduction and implementation.

Technical verification Reported

The paper contains a detailed proof, but the provenance note documents production and editing rather than a completed independent check of the model's Theorem 4. No formal or computational certificate is reported for this theorem.

Human understanding Few

Anari states he conjectured the mixing bounds but not the proof route; the manuscript explains the model-generated route, yet it does not document Anari or another human personally explaining the central square-root and entropy mechanism. Few is the evidence default.

Correction and publication history