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CS-118 · computer science · mathematics · Algorithms and data structures · Graph theory and combinatorics

Polynomial switch-chain mixing for all degree sequences

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every graphical degree sequence of a finite simple undirected n-vertex graph, does the lazy degree-preserving switch chain on all its labeled realizations mix to its uniform stationary law in polynomial time, with $t_mix\leq 2n^{8}$ and gap at least $[24n^{2}\cdot binom(n,4)]^{-1}$ whenever there is more than one realization?

Short English statement

For every graphical degree sequence of a finite simple undirected n-vertex graph, does the lazy degree-preserving switch chain on all its labeled realizations mix to its uniform stationary law in polynomial time, with $t_mix\leq 2n^{8}$ and gap at least $[24n^{2}\cdot binom(n,4)]^{-1}$ whenever there is more than one realization?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-118-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-118

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 131; 2026-09-25).

Polynomial switch-chain mixing for all degree sequences

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 131; 2026-09-25), the asserted result is that the lazy simple-undirected switch chain mixes for every graphical degree sequence in at most $2n^8$ steps at distance $1/4$, with an exact expected-polynomial sampler.

Mathematical statement of the claimed result

Manuscript 1.
Let $d$ be any graphical degree vector on $[n]$, $n\ge4$, and $\Omega_d$ all simple labeled graphs with that degree vector. The chain stays with probability $1/2$; otherwise it uniformly chooses four vertices and one of six ordered pairs of different perfect matchings on them, replacing the first by the second only when its two edges are present and both new edges absent. With uniform stationary law $\pi_d$, the claimed bounds are \[ \min\{t:\max_G\|P_d^t(G,\cdot)-\pi_d\|_{\rm TV}\le1/4\}\le2n^8, \qquad\operatorname{gap}(P_d)\ge[24n^2\binom n4]^{-1} \] when $|\Omega_d|>1$. A further claimed sampler returns an exactly uniform graph from $\Omega_d$, using unbiased bits, terminating almost surely with expected bit cost polynomial in binary encoding length of $d$.

Outline of the argument

For manuscript 1 (family 131; 2026-09-25), pair resampling operators are compared through triple and disjoint-pair variance estimates. A global operator inequality gives the spectral gap, and decomposition into individual switches transfers it to the specified lazy chain; exact residual correction yields uniform sampling.

Formal verification

For manuscript 1 (family 131; 2026-09-25), the documented Lean scope is connectivity, spectral-gap control, and the stated lazy-switch total-variation bound for graphical degree sequences; the exact uniform sampler is not part of those selected estimates.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are SwitchChain.lean.

Their named declarations include OAI.Problem315.switch_chain_main, OAI.Problem315.switch_chain_tv_bound, OAI.Problem315.switch_connectivity.

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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