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Fine-Grained Complexity Across the Computational Phase Transition — arXiv:2611.02702

Source

arXiv (en)

Original

https://arxiv.org/abs/2611.02702

License (as archived)

CC0-1.0

Archived

2026-11-03T18:48:18.981Z

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f712fb747917e8025b5e1157700fb903f4c1cc92a5539cee29c8fc32a5bffabb

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Fine-Grained Complexity Across the Computational Phase Transition — arXiv:2611.02702 — arXiv (en), https://arxiv.org/abs/2611.02702 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_f4e1069b83e2 (Metraipolis archive dc_f4e1069b83e2, sha256 f712fb747917e8025b5e1157700fb903f4c1cc92a5539cee29c8fc32a5bffabb)

@misc{dcf4e1069b83e2,
  title = {Fine-Grained Complexity Across the Computational Phase Transition — arXiv:2611.02702},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.02702},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_f4e1069b83e2; sha256 f712fb747917e8025b5e1157700fb903f4c1cc92a5539cee29c8fc32a5bffabb},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.02702. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

How does a computational problem transition from easy to hard? In the hard-core model, approximate counting and sampling are polynomial-time tractable up to tree uniqueness, but worst-case intractable at any constant relative distance above it unless $\mathsf{NP}=\mathsf{RP}$. We quantify the transition from polynomial to exponential mixing for classical Glauber dynamics. On $n$-vertex graphs of maximum degree $d\ge3$, at fugacity $\lambda=(1+\delta)\lambda_{\mathrm c}(d)$ with sufficiently small $\delta\ge0$ that may vanish with $n$, its worst-case mixing time has the form $\exp\{\Theta(n\delta^2/d)\}n^{1+\Theta(1)}$. The upper bound is degree-uniform, and the matching lower bound holds for every fixed $d$. Annealing gives approximate counting with the same running-time upper bound at fixed accuracy. For general sampling and counting algorithms, we establish a robust fine-grained reduction from SAT without assuming a lower bound on SAT's running time. For any fixed $d\ge3$ and $0<\gamma<1$, a sampling or approximate-counting algorithm running in $\exp\{O((n\delta^2/d)^{1-\gamma})\}\operatorname{poly}(n)$ time along any one suitable inverse-polynomial distance sequence $\delta=\delta(n)$, yields a randomized SAT algorithm running in $\exp\{\widetilde O(s^{1-\gamma})\}$ time on inputs of binary length $s$. This power-saving transfer identifies $n\delta^2/d$ as the analogue of SAT input length for hard-core sampling and counting at relative distance $\delta$ above criticality: beating the Glauber exponent by a fixed power through any sampling or counting method would yield a corresponding power improvement for SAT. At criticality, we establish $O(\sqrt{n/d})$ spectral independence, optimal in $n$ at fixed degree, and degree-uniform Glauber mixing in $\widetilde O(n(n/d)^{1+1/(d-2)})$ time, improving the previous $\widetilde O(n^{4+4/(d-2)})$ bound.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Computational Complexity (cs.CC) ; Data Structures and Algorithms (cs.DS); Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2611.02702 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:15:07.761Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).