The Complexity of Counting Eulerian Tours in 4-regular Graphs
The Complexity of Counting Eulerian Tours in 4-regular Graphs
复制标题
4-正则图中计算欧拉游览的复杂性
DOI：
10.1007/s00453-010-9463-4
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发表时间：
2010
期刊：
影响因子：
1.1
通讯作者：
Daniel Stefankovic
中科院分区：
文献类型：
--
作者：
Qi Ge;Daniel Stefankovic
我们从两个角度研究了计数欧拉旅行（#ET）的复杂性及其变化 - 精确计数的复杂性和复杂性W.R.T.近似估计性的降低（AP-REDUCTIONS，DYER等，Algorith，Algorithmica 38（3）（3）：471– 471– 471–471–471–471- 500，2004）我们证明#ET是＃p-complete，即使是平面4型图。 Kotzig（图理论，Proc。Colloq。，Tihany，1966年，第219-230页，学术出版社，圣地亚哥，1968年）表明，可以在多项式时间内计算＃a-trails，以用于4个规范平面图（嵌入在嵌入中平面等同于给出旋转嵌入方案）。 #ET，也就是说，我们将#ET从一般图中的#ET降低到4个规范地图中的＃A-Trails。 14（1）：274–325，2004）。
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).