{"paper":{"title":"Sum-of-Squares Lower Bounds for Independent Set in Ultra-Sparse Random Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Aaron Potechin, Jeff Xu, Pravesh Kothari","submitted_at":"2024-06-26T15:26:34Z","abstract_excerpt":"We prove that for every $D \\in \\N$, and large enough constant $d \\in \\N$, with high probability over the choice of $G \\sim G(n,d/n)$, the \\Erdos-\\Renyi random graph distribution, the canonical degree $2D$ Sum-of-Squares relaxation fails to certify that the largest independent set in $G$ is of size $o(\\frac{n}{\\sqrt{d} D^4})$. In particular, degree $D$ sum-of-squares strengthening can reduce the integrality gap of the classical \\Lovasz theta SDP relaxation by at most a $O(D^4)$ factor.\n  This is the first lower bound for $>4$-degree Sum-of-Squares (SoS) relaxation for any problems on \\emph{ultr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.18429","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.18429/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"}