{"paper":{"title":"Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Anup B. Rao, John Peebles, Jonathan Kelner, Michael B. Cohen, Rasmus Kyng, Richard Peng","submitted_at":"2018-11-26T22:33:44Z","abstract_excerpt":"We show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an $n \\times n$ Eulerian directed Laplacian with $m$ nonzero entries, we show how to compute an $\\epsilon$-approximate solution in time $O(m \\log^{O(1)} (n) \\log (1/\\epsilon))$. Through reductions from [Cohen et al. FOCS'16] , this gives the first nearly-linear time algorithms for computing $\\epsilon$-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing $\\epsilon$-approximations to various properties of random walks on dire"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10722","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"}