2026-07-21T14:19:03Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:7420 2024-03-06T10:40:28Z ddc:004 open_access

Directed Hamiltonicity and Out-Branchings via Generalized Laplacians Björklund, Andreas Kaski, Petteri Koutis, Ioannis counting directed Hamiltonicity graph Laplacian independent set k-internal out-branching 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: 1. We show that for any constant 0<lambda<1 and prime p we can count the Hamiltonian cycles modulo p^((1-lambda)n/(3p)) 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\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in O^*(3^(n-alpha(G))) time and polynomial space, where alpha(G) is the size of the maximum independent set in G. In particular, this yields an O^*(3^(n/2)) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O^*(2^k)-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andreas Björklund and Petteri Kaski and Ioannis Koutis 2017 Is Part Of LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2017.91 urn:nbn:de:0030-drops-74208 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.91 eng https://creativecommons.org/licenses/by/3.0/legalcode