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Given an input graph G=(V,E), the partition function Z_A(G) of a system is defined as\n  Z_A(G) = \\sum_{\\sigma: V -> {0,1}} \\prod_{(u,v) \\in E} A_{\\sigma(u), \\sigma(v)}\n  We prove inapproximability results for the partition function in the region specified by the non-uniqueness condition from phase transition for the Gibbs measure. More specifically, assuming NP \\ne RP, for any fixed \\beta, \\gamma in the unit square, there is no randomized polynomial-t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.2934","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-05-14T03:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"d7c8e60dc6e7f6fd253e1a386e9913b49b582097f63125eb487d740685b493a8","abstract_canon_sha256":"eb91068efeb8ba2b2a80df0fce853bfbbe915505c720977baaff171202579e46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:10.375635Z","signature_b64":"7TiwnhltmJoECflszeK7OyFyQOnO2xk2zZ+V/v+ysyl96yEPehGSFypkmgWdYdfyXzkbr4wEjGi8x5ug17k6Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"615e799184e82f58cd0bfdf121864d51862d6fffcb747cd097a6ccf067eb3849","last_reissued_at":"2026-05-18T02:21:10.375107Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:10.375107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Inapproximability After Uniqueness Phase Transition in Two-Spin Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Heng Guo, Jin-Yi Cai, Pinyan Lu, Xi Chen","submitted_at":"2012-05-14T03:10:28Z","abstract_excerpt":"A two-state spin system is specified by a 2 x 2 matrix\n  A = {A_{0,0} A_{0,1}, A_{1,0} A_{1,1}} = {\\beta 1, 1 \\gamma}\nwhere \\beta, \\gamma \\ge 0. Given an input graph G=(V,E), the partition function Z_A(G) of a system is defined as\n  Z_A(G) = \\sum_{\\sigma: V -> {0,1}} \\prod_{(u,v) \\in E} A_{\\sigma(u), \\sigma(v)}\n  We prove inapproximability results for the partition function in the region specified by the non-uniqueness condition from phase transition for the Gibbs measure. 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