{"paper":{"title":"Round and Resilience-Optimal Approximate Agreement on Trees and Block Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A synchronous protocol achieves optimal resilience for approximate agreement on trees with round complexity O(log D(T)/log log D(T)).","cross_cats":[],"primary_cat":"cs.DC","authors_text":"Diana Ghinea, Joel Rybicki, Marc Fuchs, Zahra Parsaeian","submitted_at":"2025-02-08T14:39:28Z","abstract_excerpt":"Approximate Agreement ($\\mathcal{AA}$) is a fundamental primitive that, even in the presence of Byzantine faults, allows honest parties to obtain close (but not necessarily identical) outputs that lie within the range of their inputs. While the optimal round complexity of synchronous $\\mathcal{AA}$ on real values is well understood, its extension to other input spaces has remained open, with fundamental questions regarding achievable resilience and round efficiency still unresolved. In this work, we investigate the optimal round complexity of synchronous $\\mathcal{AA}$ on trees under Byzantine"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a synchronous protocol with optimal resilience and round complexity O(log D(T)/log log D(T)), where D(T) denotes the diameter of the input space tree. Complementing this result, we extend impossibility results for real-valued AA to any graph G by proving a lower bound of Omega(log D(G)/log log D(G) + log (n+t)/t) rounds. Together, these results establish the asymptotic optimality of our protocol whenever t in Theta(n).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The impossibility results for real-valued AA can be extended to arbitrary graphs G while preserving the convex hull and 1-close output requirements (abstract, paragraph on lower bound extension).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Optimal round complexity and resilience approximate agreement on trees with matching lower bounds, extended to block graphs in synchronous and asynchronous models.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A synchronous protocol achieves optimal resilience for approximate agreement on trees with round complexity O(log D(T)/log log D(T)).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5a9a7393e6ca2c034861b64888e27340a35289437b4993311a7ed128f2217a70"},"source":{"id":"2502.05591","kind":"arxiv","version":5},"verdict":{"id":"959f5159-ebe3-4670-9391-357dd0c3b72f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T04:01:26.701207Z","strongest_claim":"We present a synchronous protocol with optimal resilience and round complexity O(log D(T)/log log D(T)), where D(T) denotes the diameter of the input space tree. Complementing this result, we extend impossibility results for real-valued AA to any graph G by proving a lower bound of Omega(log D(G)/log log D(G) + log (n+t)/t) rounds. Together, these results establish the asymptotic optimality of our protocol whenever t in Theta(n).","one_line_summary":"Optimal round complexity and resilience approximate agreement on trees with matching lower bounds, extended to block graphs in synchronous and asynchronous models.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The impossibility results for real-valued AA can be extended to arbitrary graphs G while preserving the convex hull and 1-close output requirements (abstract, paragraph on lower bound extension).","pith_extraction_headline":"A synchronous protocol achieves optimal resilience for approximate agreement on trees with round complexity O(log D(T)/log log D(T))."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.05591/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}