Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed $\epsilon >0$, unless $\mathrm{P}=\mathrm{NP}$, there exists no polynomial-time $n^{1-\epsilon }$-approximation algorithm even for bipartite planar graphs. We then give an $\alpha (\mathrm{\Delta }-1)/2$-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in $t(\alpha ,G)+O(\mathrm{\Delta }n)$ time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in $t(\alpha ,G)$ time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

给定具有n个顶点的无向图G，独立反馈顶点集问题是找到顶点数量最少的G的顶点子集F，使得F既是独立集又是G的反馈顶点集（如果存在）。对于最大四度的二分平面图，已知此问题是NP难的。在本文中，我们研究了问题的逼近性。我们首先证明，对于任何固定$\epsilon >0$，除非 $\mathrm{P}=\mathrm{NP}$，不存在多项式时间 ${ñ}^{\mathrm{1个}-\epsilon }$-近似算法，甚至适用于二部平面图。然后，我们给$\alpha （\mathrm{\Delta }-\mathrm{1个}）/2$最大度为Δ的二部图G的近似逼近算法$Ť（\alpha ，G）+Ø（\mathrm{\Delta }ñ）$时间，假设在二部图上原始反馈顶点集问题有一个α近似算法，该算法在$Ť（\alpha ，G）$时间。该算法结果还产生了针对最大三度二部图中的独立反馈顶点集问题的多项式时间（精确）算法。