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Faster exponential-time algorithms in graphs of bounded average degree

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Cygan, Marek and Pilipczuk, Marcin (2015) Faster exponential-time algorithms in graphs of bounded average degree. Information and Computation, 243 . pp. 75-85. doi:10.1016/j.ic.2014.12.007 ISSN 0890-5401.

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Official URL: http://dx.doi.org/10.1016/j.ic.2014.12.007

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Abstract

We present a number of exponential-time algorithms for problems in sparse matrices and graphs of bounded average degree. First, we obtain a simple algorithm that computes a permanent of an n×n matrix over an arbitrary commutative ring with at most dn non-zero entries using O⋆(2(1−1/(3.55d))n) time and ring operations, 1 improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].

Second, we present a simple algorithm for counting perfect matchings in an n -vertex graph in O⋆(2n/2) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Björklund [SODA 2012], but relies on inclusion–exclusion principle instead of algebraic transformations.

Third, we show a combinatorial lemma that bounds the number of “Hamiltonian-like” structures in a graph of bounded average degree. Using this result, we show that

1. a minimum weight Hamiltonian cycle in an n-vertex graph with average degree bounded by d can be found in O⋆(2(1−εd)n) time and exponential space for a constant εd depending only on d;
2. the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in View the MathML source time and exponential space, for a constant View the MathML source depending only on d.

The algorithm for minimum weight Hamiltonian cycle generalizes the recent results of Björklund et al. [TALG 2012] on graphs of bounded degree.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Library of Congress Subject Headings (LCSH): Computer algorithms, Hamiltonian systems
Journal or Publication Title: Information and Computation
Publisher: Academic Press
ISSN: 0890-5401
Official Date: August 2015
Dates:
DateEvent
August 2015Published
11 December 2014Available
15 August 2013Published
Volume: 243
Number of Pages: 11
Page Range: pp. 75-85
DOI: 10.1016/j.ic.2014.12.007
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 28 December 2015
Date of first compliant Open Access: 28 December 2015
Funder: Narodowe Centrum Nauki (NCN), Fundacja na rzecz Nauki Polskiej [Foundation for Polish Science] (FNP)
Grant number: N206567140 (NCN)

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