极大极小FVS的（不）逼近性
(In)approximability of Maximum Minimal FVS
摘要：
研究了NP完全\textsc{Maximum Minimal Feedback Vertex Set}问题的可逼近性。非正式地说，这个自然问题似乎位于两个更好研究的这类问题之间的中间空间：\textsc{Maximum Minimal Vertex Cover}，其中最佳可实现的近似比是$\sqrt{n}$，以及\textsc{Upper Dominating Set}，其中不允许任何$n^{1-ε}$近似。我们确认和量化这种直觉，显示了第一个非平凡的多项式时间近似\textsc{Max Min FVS}的比例为$O(n^{2/3})$，以及匹配的硬度近似界的$n^{2/3-ε}$，提高了以前已知的硬度$n^{1/2-ε}$。近似算法也给出了一个三次内核时，参数化的解决方案的大小。一路上，我们还获得了$O(Δ)$ -近似，并表明这是渐近最好的可能，我们提高了该问题是NP-难从$Δ\ge 9$到$Δ\ge 6$的界限。在解决了多项式时间内问题的可逼近性之后，我们转向超多项式时间的上下文中.我们设计了一个概括我们的近似算法，对于任何所需的近似比$r$，产生一个$r$ -近似的解决方案，在时间$n^{O(n/r^{3/2})}$。这种时间近似的权衡是本质上紧：我们表明，在ETH下，对于任何比例$r$和$ε>0$，没有算法可以$r$ -近似这个问题的时间$n^{O((n/r^{3/2})^{1-ε})}$，因此，我们精确地描述了多项式和次指数时间之间的整个频谱的问题的可近似性，直到第二个指数中的任意小常数。
英文摘要：
We study the approximability of the NP-complete \textsc{Maximum Minimal Feedback Vertex Set} problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: \textsc{Maximum Minimal Vertex Cover}, for which the best achievable approximation ratio is $\sqrt{n}$, and \textsc{Upper Dominating Set}, which does not admit any $n^{1-ε}$ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for \textsc{Max Min FVS} with a ratio of $O(n^{2/3})$, as well as a matching hardness of approximation bound of $n^{2/3-ε}$, improving the previous known hardness of $n^{1/2-ε}$. The approximation algorithm also gives a cubic kernel when parameterized by the solution size. Along the way, we also obtain an $O(Δ)$-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from $Δ\ge 9$ to $Δ\ge 6$. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio $r$, produces an $r$-approximate solution in time $n^{O(n/r^{3/2})}$. This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio $r$ and $ε>0$, no algorithm can $r$-approximate this problem in time $n^{O((n/r^{3/2})^{1-ε})}$, hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.
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