在单纯复形中求2-球面的参数化复杂性
The parameterized complexity of finding a 2-sphere in a simplicial complex
摘要：
我们考虑寻找单纯复形K的子复形K'，使得K'同胚于二维球面S^2的问题。我们研究这个问题的两个变种。第一个问题是是否存在这样一个K'，最多有k个三角形，我们证明了这个变体是W[1]-hard的，并且假设ETH，不允许O（n^{o（sqrt（k））}）时间算法。我们还给出了一个算法，是紧关于这个下界。第二个问题是第一个问题的对偶问题，它询问是否可以通过从K中删除最多k个三角形来找到K'。这个变体有一个即时的O（3^k poly（|K|））时间算法，我们证明了它允许一个多项式核化到O（k^2）三角形，以及一个多项式压缩到位大小为O（k log k）的加权版本。
英文摘要：
We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^{o(sqrt(k))}) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).
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