Abstract
We are motivated by a tantalizing open question in exact algorithms: can we detect whether an nvertex 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^((1lambda)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^(nalpha(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 exponentialspace 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 Laplacianlike matrices, inspired by the MatrixTree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the MatrixTree Theorem to derive simple algebraic algorithms for detecting outbranchings. Specifically, we give an O^*(2^k)time randomized algorithm for detecting outbranchings 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 kLeaf problem, based on a nonstandard monomial detection problem.
BibTeX  Entry
@InProceedings{bjrklund_et_al:LIPIcs:2017:7420,
author = {Andreas Bj{\"o}rklund and Petteri Kaski and Ioannis Koutis},
title = {{Directed Hamiltonicity and OutBranchings via Generalized Laplacians}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {91:191:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7420},
URN = {urn:nbn:de:0030drops74208},
doi = {10.4230/LIPIcs.ICALP.2017.91},
annote = {Keywords: counting, directed Hamiltonicity, graph Laplacian, independent set, kinternal outbranching}
}
Keywords: 

counting, directed Hamiltonicity, graph Laplacian, independent set, kinternal outbranching 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 