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DOI: 10.4230/LIPIcs.ICALP.2017.91
URN: urn:nbn:de:0030-drops-74208
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Björklund, Andreas ; Kaski, Petteri ; Koutis, Ioannis

Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

LIPIcs-ICALP-2017-91.pdf (0.6 MB)


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.

BibTeX - Entry

  author =	{Andreas Bj{\"o}rklund and Petteri Kaski and Ioannis Koutis},
  title =	{{Directed Hamiltonicity and Out-Branchings via Generalized Laplacians}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{91:1--91:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-74208},
  doi =		{10.4230/LIPIcs.ICALP.2017.91},
  annote =	{Keywords: counting, directed Hamiltonicity, graph Laplacian, independent set, k-internal out-branching}

Keywords: counting, directed Hamiltonicity, graph Laplacian, independent set, k-internal out-branching
Collection: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 07.07.2017

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