{"paper":{"title":"Directed Hamiltonicity and Out-Branchings via Generalized Laplacians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Andreas Bj\\\"orklund, Ioannis Koutis, Petteri Kaski","submitted_at":"2016-07-14T06:06:39Z","abstract_excerpt":"We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve upon several previous works:\n  1. We show that for any constant $0<\\lambda<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p^{\\lfloor (1-\\lambda)\\frac{n}{3p}\\rfloor}$ in expected time less than $c^n$ for a constant $c<2$ that depends only on $p$ and $\\lambda$. Such an algorithm was previously known only for the case of counting modulo two [Bj\\\"orkl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04002","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"}