{"paper":{"title":"Faster spectral sparsification and numerical algorithms for SDD matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Alex Levin, Ioannis Koutis, Richard Peng","submitted_at":"2012-09-26T03:15:16Z","abstract_excerpt":"We study algorithms for spectral graph sparsification. The input is a graph $G$ with $n$ vertices and $m$ edges, and the output is a sparse graph $\\tilde{G}$ that approximates $G$ in an algebraic sense. Concretely, for all vectors $x$ and any $\\epsilon>0$, $\\tilde{G}$ satisfies $$ (1-\\epsilon) x^T L_G x \\leq x^T L_{\\tilde{G}} x \\leq (1+\\epsilon) x^T L_G x, $$ where $L_G$ and $L_{\\tilde{G}}$ are the Laplacians of $G$ and $\\tilde{G}$ respectively. We show that the fastest known algorithm for computing a sparsifier with $O(n\\log n/\\epsilon^2)$ edges can actually run in $\\tilde{O}(m\\log^2 n)$ time"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5821","kind":"arxiv","version":3},"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"}