{"paper":{"title":"On sub-determinants and the diameter of polyhedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Friedrich Eisenbrand, Marco Di Summa, Martin Niemeier, Nicolai H\\\"ahnle, Nicolas Bonifas","submitted_at":"2011-08-22T10:23:47Z","abstract_excerpt":"We derive a new upper bound on the diameter of a polyhedron P = {x \\in R^n : Ax <= b}, where A \\in Z^{m\\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \\Delta. More precisely, we show that the diameter of P is bounded by O(\\Delta^2 n^4 log n\\Delta). If P is bounded, then we show that the diameter of P is at most O(\\Delta^2 n^3.5 log n\\Delta).\n  For the special case in which A is a totally unimodular matrix, the bounds are O(n^4 log n) and O(n^3.5 log n) respectively. This improves over the previous best bound of O(m^16 n^3 (log mn)^3) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4272","kind":"arxiv","version":2},"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"}