{"paper":{"title":"Bounding the Bethe and the Degree-$M$ Bethe Permanents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math-ph","math.CO","math.IT","math.MP"],"primary_cat":"cs.IT","authors_text":"Martin Haenggi, Roxana Smarandache","submitted_at":"2015-03-07T21:55:43Z","abstract_excerpt":"It was recently conjectured that the permanent of a ${P}$-lifting $\\theta^{\\uparrow{P}}$ of a matrix $\\theta$ of degree $M$ is less than or equal to the $M$th power of the permanent perm$(\\theta)$, i.e., perm$(\\theta^{\\uparrow{P}})\\leq(\\text{perm}(\\theta))^M$ and, consequently, that the degree-$M$ Bethe permanent $\\text{perm}_{M,\\mathrm{B}} (\\theta)$ of a matrix $\\theta$ is less than or equal to the permanent perm$(\\theta)$ of $\\theta$, i.e., perm$_{M, \\mathrm{B}} (\\theta)\\leq \\text{perm}(\\theta)$. In this paper, we prove these related conjectures and show in addition a few properties of the p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02217","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}