{"paper":{"title":"Simultaneously Satisfying Linear Equations Over $\\mathbb{F}_2$: MaxLin2 and Max-$r$-Lin2 Parameterized Above Average","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"A. Yeo, F. Rosamond, G. Gutin, M. Fellows, M. Jones, R. Crowston, S. Thomasse","submitted_at":"2011-04-06T16:11:03Z","abstract_excerpt":"In the parameterized problem \\textsc{MaxLin2-AA}[$k$], we are given a system with variables $x_1,...,x_n$ consisting of equations of the form $\\prod_{i \\in I}x_i = b$, where $x_i,b \\in \\{-1, 1\\}$ and $I\\subseteq [n],$ each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least $W/2+k$, where $W$ is the total weight of all equations and $k$ is the parameter (if $k=0$, the possibility is assured). We show that \\textsc{MaxLin2-AA}[$k$] has a kernel with at most $O(k^2\\log k)$ variables and can be solved in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1135","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"}