求有界树宽单纯复形曲面的ETH-紧算法
ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth
摘要：
给定一个具有$n$单形的单纯复形，我们考虑连通子曲面识别（c-SR）问题，即找到一个与给定的具有固定边界的连通曲面同胚的子复形。我们还研究了相关的总和属次表面识别（SoG）问题，在那里我们不是寻找一个表面的边界，连接组件的数量，和总属。对于这两个问题，我们给参数化算法的树宽$k$的Hasse图，运行在$2^{O(k \log k)}n^{O(1)}$时间。对于SoG问题，我们还证明了我们的算法是最优的假设指数时间的假设。事实上，我们证明了更强的结果，我们的算法是ETH紧的，即使没有限制的总亏格。
英文摘要：
Given a simplicial complex with $n$ simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth $k$ of the Hasse diagram that run in $2^{O(k \log k)}n^{O(1)}$ time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.
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