横截秩、共形性与计数
Transversal Rank, Conformality and Enumeration
AI总结：
本文提出了一种识别横截秩至少为k的超图的算法，并探讨了通过改进运行时间来提升计算效率的可能性。
摘要：
超图的横截秩是其最小击中集的最大大小。对于一个具有n个顶点、m条边的超图和一个整数k，判断其横截秩是否至少为k的时间复杂度为O(m^{k+1}n)，这已知的算法可以追溯到70年代。它基本上与Araújo、Bougeret、Campos和Sau[Algorithmica 2023]以及Dublois、Lampis和Paschos[TCS 2022]提出的( m + n )^{Ω(k)} ETH下界相匹配。许多实际中出现的超图具有比顶点更多的边，m >> n。这引发了疑问，是否可以改进运行时间对m的依赖性，以换取对n的依赖性增加。我们的第一个结果是，识别横截秩至少为k的超图的时间复杂度为O(Δ^{k-2} mn^{k-1})，其中Δ ≤ m是最大度数。我们的主要技术贡献是“前瞻”方法，它允许我们找到更高阶的扩展，即最小击中集，这些集可以将给定的集合增加至少两个顶点。我们证明，这种方法也可以用于枚举超图的所有最小击中集，延迟为O(Δ^{k*-1} mn^2)。然后，我们探讨了进一步减少计算横截秩运行时间的可能性，即为poly(m)·n^{k+O(1)}。这实际上等同于组合算法和枚举中的几个突破。其他方面，这种改进只有在k-共形超图可以以poly(m)·n^{k+O(1)}的时间内被识别，并且在均匀超图中可以以增量延迟枚举最大超 clique/独立集的情况下才有可能。
英文摘要：
The transversal rank of a hypergraph is the maximum size of its minimal hitting sets. Deciding, for an $n$-vertex, $m$-edge hypergraph and an integer $k$, whether the transversal rank is at least $k$ takes time $O(m^{k+1} n)$ with an algorithm that is known since the 70s. It essentially matches an $(m+n)^{Ω(k)}$ ETH-lower bound by Araújo, Bougeret, Campos, and Sau [Algorithmica 2023] and Dublois, Lampis, and Paschos [TCS 2022]. Many hypergraphs seen in practice have much more edges than vertices, $m \gg n$. This raises the question whether an improvement of the run time dependency on $m$ can be traded for an increase in the dependency on $n$. Our first result is an algorithm to recognize hypergraphs with transversal rank at least $k$ in time $O(Δ^{k-2} mn^{k-1})$, where $Δ\le m$ is the maximum degree. Our main technical contribution is a ``look-ahead'' method that allows us to find higher-order extensions, minimal hitting sets that augment a given set with at least two more vertices. We show that this method can also be used to enumerate all minimal hitting sets of a hypergraph with transversal rank $k^*$ with delay $O(Δ^{k^*-1} mn^2)$. We then explore the possibility of further reducing the running time for computing the transversal rank to $\textsf{poly}(m) \cdot n^{k+O(1)}$. This turns out to be equivalent to several breakthroughs in combinatorial algorithms and enumeration. Among other things, such an improvement is possible if and only if $k$-conformal hypergraphs can also be recognized in time $\textsf{poly}(m) \cdot n^{k+O(1)}$, and iff the maximal hypercliques/independent sets of a uniform hypergraph can be enumerated with incremental delay.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
