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Structural Corrections to the Bethe Approximation of the Permanent — arXiv:2608.31061

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.31061
License (as archived): CC0-1.0
Archived: 2026-10-17T09:31:24.387Z
SHA-256 of the archived text: 60bd105a046e24706e125677025568a9b289196735e40eb87a983a27065b64f9
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How to cite

Structural Corrections to the Bethe Approximation of the Permanent — arXiv:2608.31061 — arXiv (en), https://arxiv.org/abs/2608.31061 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_514b6a326893 (Metraipolis archive dc_514b6a326893, sha256 60bd105a046e24706e125677025568a9b289196735e40eb87a983a27065b64f9)

BibTeX

@misc{dc514b6a326893,
  title = {Structural Corrections to the Bethe Approximation of the Permanent — arXiv:2608.31061},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.31061},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_514b6a326893; sha256 60bd105a046e24706e125677025568a9b289196735e40eb87a983a27065b64f9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.31061. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-01 in math.CO).
Categorie arXiv: Data Structures and Algorithms (cs.DS) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.31061 — 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 2026-10-17T09:02:28.892Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).