Abstract
We investigate the complexity of counting Eulerian tours (#ET) and its variations from two perspectives—the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions, Dyer et al., Algorithmica 38(3):471–500, 2004). We prove that #ET is #P-complete even for planar 4-regular graphs.
A closely related problem is that of counting A-trails (#A-trails) in graphs with rotational embedding schemes (so called maps). Kotzig (Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 219–230, Academic Press, San Diego, 1968) showed that #A-trails can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivalent to giving a rotational embedding scheme). We show that for 4-regular maps the problem is #P-hard. Moreover, we show that from the approximation viewpoint #A-trails in 4-regular maps captures the essence of #ET, that is, we give an AP-reduction from #ET in general graphs to #A-trails in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem (Wilson, Ann. Appl. Probab. 14(1):274–325, 2004).
Similar content being viewed by others
References
Andersen, L.D., Fleischner, H.: The NP-completeness of finding A-trails in Eulerian graphs and of finding spanning trees in hypergraphs. Discrete Appl. Math. 59(3), 203–214 (1995)
Bach, E., Shallit, J.: Algorithmic Number Theory. Vol. 1. Foundations of Computing Series. MIT Press, Cambridge (1996)
Bent, S.W., Manber, U.: On nonintersecting Eulerian circuits. Discrete Appl. Math. 18(1), 87–94 (1987)
Brightwell, G., Winkler, P.: Counting Eulerian circuits is #P-complete. In: ALENEX/ANALCO, pp. 259–262 (2005)
Creed, P.: Counting and sampling problems on Eulerian graphs. Submitted Ph.D. Dissertation, University of Edinburgh (2010)
Dvořák, Z.: Eulerian tours in graphs with forbidden transitions and bounded degree. KAM-DIMATIA (669) (2004)
Dyer, M., Goldberg, L.A., Greenhill, C., Jerrum, M.: The relative complexity of approximate counting problems. Algorithmica 38(3), 471–500 (2004)
Fleischner, H.: Eulerian Graphs and Related Topics. Part 1. Vol. 1. Annals of Discrete Mathematics, vol. 45. North-Holland, Amsterdam (1990)
Ge, Q., Štefankovič, D.: The complexity of counting Eulerian tours in 4-regular graphs. In: LATIN, pp. 638–649 (2010)
Ge, Q., Štefankovič, D.: The complexity of counting Eulerian tours in 4-regular graphs. arXiv:1009.5019 (2010)
Jerrum, M.: Review MR1822924 (2002k:68197) of [14]. MathSciNet (2002)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2003)
Kotzig, A.: Eulerian lines in finite 4-valent graphs and their transformations. In: Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 219–230. Academic Press, San Diego (1968)
Tetali, P., Vempala, S.: Random sampling of Euler tours. Algorithmica 30(3), 376–385 (2001)
Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)
Wilson, D.B.: Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14(1), 274–325 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported, in part, by NSF grant CCF-0910584. This paper is an extension of the previous work presented at the 9th Latin American Theoretical Informatics Symposium (LATIN 2010) [9].
Rights and permissions
About this article
Cite this article
Ge, Q., Štefankovič, D. The Complexity of Counting Eulerian Tours in 4-regular Graphs. Algorithmica 63, 588–601 (2012). https://doi.org/10.1007/s00453-010-9463-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-010-9463-4