We study the Independent Feedback Vertex Set problem — a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set \(S\subseteq V(G)\) such that G ∖ S is a forest and S is an independent set of size at most k. We present an \(\mathcal {O}^{\ast }((1+\varphi ^{2})^{k})\)-time FPT algorithm for this problem, where φ < 1.619 is the golden ratio, improving the previous fastest \(\mathcal {O}^{\ast }(4.1481^{k})\)-time algorithm given by Agrawal et al. (2016). The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem. On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm.

我们研究了独立反馈顶点集问题-经典反馈顶点集问题的变体，其中给定一个图G和一个整数k，问题是要确定是否存在一个顶点集\（S \ subseteq V（G）\ ），使得G ∖ S是森林，S是最多k的独立大小集。针对此问题，我们提出了\（\ mathcal {O} ^ {\ ast}（（1+ \ varphi ^ {2}）^ {k}）\） -时间FPT算法，其中φ <1.619是黄金分割率，改进以前最快的\（\ mathcal {O} ^ {\ ast}（4.1481 ^ {k}）\）Agrawal等人给出的实时算法。（2016）。我们的时间复杂度范围内的指数因子与经典反馈顶点集问题的最快确定性FPT算法匹配。在技​​术方面，主要的新颖之处在于在分支过程中对输入实例的精确度量，从而可以对算法进行更简单，更简洁的描述和分析。