The complexity of counting eulerian tours in 4-regular graphs

Authors:
Qi Ge;Daniel Štefankovič
Affiliations:
Department of Computer Science, University of Rochester;Department of Computer Science, University of Rochester
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

Citing 4
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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 [DGGJ04]). 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 [Kot68] 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 [Wil04].