Approximability of the independent feedback vertex set problem for bipartite graphs☆
Introduction
A feedback vertex set F of an undirected graph is a vertex subset of G such that the subgraph of G induced by is a forest. (See Fig. 1(b) as an example.) For a given graph G, the feedback vertex set problem is to find a feedback vertex set of G with the minimum number of vertices. The feedback vertex set problem is one of the most classical NP-hard problems, and many algorithms have been developed from various viewpoints over the years.
Misra et al. [16] introduced an independence variant of the feedback vertex set problem. An independent set I of a graph is a vertex subset of G such that the subgraph of G induced by I contains no edge. A vertex set of G is said to be an independent feedback vertex set of G if it is both an independent set and a feedback vertex set of G. (See Fig. 1(c).) Note that not every graph has an independent feedback vertex set; for example, consider a complete graph with four or more vertices. For a given graph G, the independent feedback vertex set problem is to find an independent feedback vertex set of G with the minimum number of vertices, if it exists. For convenience, we sometimes call the feedback vertex set problem the original problem, and the independent feedback vertex set problem the independence variant.
The original problem is NP-hard even for bipartite planar graphs of maximum degree four [17], while it can be solved in polynomial time for any graph of maximum degree at most three [18], [21]. Furthermore, the original problem is APX-complete for bipartite graphs of maximum degree at least six.4 This means that the original problem is unlikely to have a polynomial-time approximation scheme (PTAS). The original problem has been intensively studied from various viewpoints, such as of approximation [3], [4], [6], fixed-parameter tractability [11], [13], and tractability on special graph classes [10], [12], [18].
As Misra et al. pointed out in [16], by inserting a new vertex in every edge of a graph, the original problem can be reduced to the independence variant without changing the size of optimal solutions. This implies that the independence variant is NP-hard even for bipartite planar graphs of maximum degree four, and is APX-hard for bipartite graphs of maximum degree at least six. In the same paper, Misra et al. also developed a fixed-parameter algorithm whose running time is , where k is the solution size as the parameter. This running time was improved to by Agrawal et al. [1], and then to by Li and Pilipczuk [14].
The independence variant has been also studied from the viewpoint of graph classes. For example, it is solvable in polynomial time for bounded treewidth graphs [19], chordal graphs [19], -free graphs [9], and graphs of diameter two [8]. Interestingly, for the latter two graph classes, their polynomial-time solvabilities of the original problem remain open.
The independence variant is strongly related to the near-bipartiteness problem [7], [9], [22]. In the problem, for a given graph G, our task is to decide whether G has at least one independent feedback vertex set. Therefore, the intractability of the independence variant is inherited from the near-bipartiteness problem. The near-bipartiteness problem is known to be NP-complete even for graphs of maximum degree four [22], graphs of diameter at most three [7], and for line graphs of bipartite planar subcubic graphs [9]. Note that, since any bipartite graph has an independent feedback vertex set, the near-bipartiteness problem is trivially solvable for bipartite graphs.
In this paper, we study the approximability of the independent feedback vertex set problem for bipartite graphs. As we will see in Section 2, we can safely discuss the approximation factor of this (minimization) problem for bipartite graphs. Recall that the independence variant is APX-hard for bipartite graphs.
We first give a stronger inapproximability result. We show that, unless , the independence variant admits no polynomial-time approximation algorithm within a factor for any fixed , even on bipartite planar graphs, where n is the number of vertices in a graph. This gives a contrast to the existence of polynomial-time 2-approximation algorithms for the original problem on general graphs [4], [6]. One might think that this hardness result is straightforward, because the independence variant has the constraint of independent sets. In fact, the independent set problem, which finds a maximum-size independent set of a given graph, is also hard to approximate within a factor in polynomial time, for any fixed [23]. However, since the independent set problem is a maximization problem whereas the independent feedback vertex set problem is a minimization problem, it is not so straightforward to give our hardness result. We also point out that the independent set problem admits a PTAS for planar graphs [5], and is solvable in polynomial time for bipartite graphs (from König's theorem).
We then give an -approximation algorithm for bipartite graphs G of n vertices and maximum degree Δ, which runs in time, under the assumption that there is an α-approximation algorithm for the original problem on bipartite graphs which runs in time. As we will show later, this result yields a polynomial-time -approximation algorithm for the independence variant on bipartite graphs. In contrast, our inapproximability result says that, unless , there is no polynomial-time -approximation algorithm for any fixed . Thus, our approximation factor is best possible with respect to the exponent of Δ.
We finally note that our algorithmic theorem yields interesting contrasts of complexity and approximability of the independent variant on bipartite graphs with respect to maximum degree. For example, we can obtain a polynomial-time (exact) algorithm for the independent variant on bipartite graphs of maximum degree three, whereas the problem is NP-hard for bipartite planar graphs of maximum degree four. We will show more results in Section 4.2, and summarize them in Table 1.
Section snippets
Preliminaries
In this paper, we assume that graphs are undirected, unweighted, and simple. We can assume that graphs are connected without loss of generality. Let be a graph; we sometimes denote by and the vertex set and edge set of G, respectively. For a vertex v in G, we denote by the set of all neighbors of v in G, that is, . For a vertex subset of a graph , let be the subgraph of G induced by . For a subset , we denote simply by the
Inapproximability
As mentioned in Introduction, the independent feedback vertex set problem is APX-hard even for bipartite graphs. In this section, we give the following stronger result. Theorem 1 Let be any fixed constant. The independent feedback vertex set problem admits no polynomial-time approximation algorithm within a factor for bipartite planar graphs of n vertices, unless .
Approximation algorithm
In this section, we deal with any bipartite graph of maximum degree , which is not necessarily planar. Note that the independent feedback vertex set problem is solvable in linear time for any graph of maximum degree at most two, because each connected component of such a graph is either a path or a cycle. In this section, we give the following theorem. Theorem 2 Let G be a bipartite graph of n vertices and maximum degree . Suppose that, for the feedback vertex set problem, there is an
Conclusions
In this paper, we analyzed the approximability of the independent feedback vertex set problem for bipartite graphs, and gave several interesting contrasts with respect to maximum degree Δ, as shown in Table 1. We also note that our approximation factors in Corollary 2, Corollary 3 are best possible with respect to the exponent of Δ.
It remains open to clarify whether there exists a PTAS for the independent feedback vertex set problem on bipartite graphs of maximum degree .
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We are grateful to anonymous referees of the preliminary version [20] and of this journal version for their helpful suggestions. We thank the support by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University, when Yuma Tamura presented the preliminary version of this paper.
References (23)
- et al.
Some APX-completeness results for cubic graphs
Theor. Comput. Sci.
(2000) - et al.
Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem
Artif. Intell.
(1996) - et al.
Independent feedback vertex sets for graphs of bounded diameter
Inf. Process. Lett.
(2018) - et al.
Faster deterministic feedback vertex set
Inf. Process. Lett.
(2014) - et al.
On parameterized independent feedback vertex set
Theor. Comput. Sci.
(2012) - et al.
On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three
Discrete Math.
(1988) - et al.
Partition the vertices of a graph into one independent set and one acyclic set
Discrete Math.
(2006) - et al.
Improved algorithms and combinatorial bounds for independent feedback vertex set
- et al.
Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs
- et al.
A 2-approximation algorithm for the undirected feedback vertex set problem
SIAM J. Discrete Math.
(1999)
Approximation algorithms for NP-complete problems on planar graphs
J. ACM
Cited by (0)
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A preliminary version has appeared in proceedings of the 14th International Conference and Workshop on Algorithms and Computation (WALCOM 2020) [20].
- 1
Partially supported by JSPS KAKENHI Grant Number JP20J11259, Japan.
- 2
Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan.
- 3
Partially supported by JSPS KAKENHI Grant Number JP19K11813, Japan.