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- W2952626921 abstract "We investigate the complexity of counting Eulerian tours ({sc #ET}) and its variations from two perspectives---the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions cite{MR2044886}). We prove that {sc #ET} is #P-complete even for planar 4-regular graphs. A closely related problem is that of counting A-trails ({sc #A-trails}) in graphs with rotational embedding schemes (so called maps). Kotzig cite{MR0248043} showed that {sc #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 {sc #A-trails} in 4-regular maps captures the essence of {sc #ET}, that is, we give an AP-reduction from {sc #ET} in general graphs to {sc #A-trails} in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem cite{MR2023023}. In order to understand whether #{sc A-trails} in 4-regular maps can AP-reduce to #{sc ET} in 4-regular graphs, we investigate a problem in which transitions in vertices are weighted (this generalizes both #{sc A-trails} and #{sc ET}). In the 4-regular case we show that {sc A-trails} can be used to simulate any vertex weights and provide evidence that {sc ET} can simulate only a limited set of vertex weights." @default.
- W2952626921 created "2019-06-27" @default.
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- W2952626921 date "2010-09-25" @default.
- W2952626921 modified "2023-09-26" @default.
- W2952626921 title "The Complexity of Counting Eulerian Tours in 4-Regular Graphs" @default.
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- W2952626921 doi "https://doi.org/10.48550/arxiv.1009.5019" @default.
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