Approximability of the Independent Feedback Vertex Set Problem for Bipartite Graphs

Given a 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. In this paper, we study the approximability of the problem. We first show that, for any fixed \(\varepsilon > 0\), unless \(\mathrm{P} = \mathrm{NP}\), there exists no polynomial-time \(n^{1-\varepsilon }\)-approximation algorithm even for bipartite planar graphs. This gives a contrast to the existence of a polynomial-time 2-approximation algorithm for the original feedback vertex set problem on general graphs. We then give an \(\alpha (\mathrm{\Delta }-1)/2\)-approximation algorithm for bipartite graphs G of maximum degree \(\mathrm{\Delta }\), which runs in \(O(t(G)+\mathrm{\Delta }n)\) time, under the assumption that there is an \(\alpha \)-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in O(t(G)) time.