Approximability of the Independent Feedback Vertex Set Problem for Bipartite Graphs

Abstract

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.

T. Ito—Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan.

X. Zhou—Partially supported by JSPS KAKENHI Grant Number JP19K11813, Japan.