Parameterized complexity of fair feedback vertex set problem

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Abstract

Given a graph G=(V,E), a subset SV(G) is said to be a feedback vertex set of G if GS is a forest. In the Feedback Vertex Set (FVS) problem, we are given an undirected graph G, and a positive integer k, the question is whether there exists a feedback vertex set of size at most k. In this paper, we study three variants of the FVS problem: Unrestricted Fair FVS, Restricted Fair FVS, and Relaxed Fair FVS. In Unrestricted Fair FVS, we are given a graph G and a positive integer , the question is does there exist a feedback vertex set SV(G) (of any size) such that for every vertex vV(G), v has at most neighbours in S. First, we study Unrestricted Fair FVS from different parameterizations such as treewidth, treedepth, and neighbourhood diversity and obtain several results (both tractability and intractability). Next, we study Restricted Fair FVS, where we are also given an integer k in the input and we demand the size of S to be at most k. This problem is trivially NP-complete; we show that Restricted Fair FVS when parameterized by the solution size k and the maximum degree Δ of the graph G, admits a kernel of size O(Δk). Finally, we study the Relaxed Fair FVS problem, where we want that the size of S is at most k and for every vertex v outside S, v has at most neighbours in S. We give an FPT algorithm for Relaxed Fair FVS problem running in time cknO(1), for a fixed constant c.

Introduction

Feedback Vertex Set (FVS) problem is one of Karp's 21 NP-complete problems [9]. This problem has been studied extensively in computational complexity theory, as it is one of the fundamental problems in the theory. The FVS problem has applications in operating systems, database systems, and VLSI design. It is used for resource allocation in operating systems. In the system resource allocation graph, the FVS problem is used for deadlock recovery [17].

FVS is a vertex deletion problem which demands to find a set SV(G) of size at most k such that the remaining graph GS is a forest. The set SV(G) is called a feedback vertex set of G. For a vertex deletion problem, given a graph G=(V,E) and property Π, we are asked to find a set SV such that the subgraph obtained after the deletion of vertices in S, that is, the graph GS satisfies the desired property Π. In the optimization version of deletion problems (vertex deletion/edge deletion), we focus on minimizing the total number of elements required to remove to achieve the desired property on the remaining graph. Modified versions of deletion problems, called fair deletion problems were introduced by Lin and Sahni in 1989 [15]. Unlike usual deletion problems, fair deletion problems aim to minimize the maximum number of neighbours contributed to the solution set by a single vertex in the graph. Given a set SV(G), the fair cost of S is maxvV{|N(v)S|}. In this paper, we study the parameterized complexity of three variants of FVS, namely, Unrestricted Fair Feedback Vertex Set (Unrestricted Fair FVS), Restricted Fair Feedback Vertex Set (Restricted Fair FVS), and Relaxed Fair Feedback Vertex Set (Relaxed Fair FVS). We define these variants as follows:

A problem with input size n and parameter k is said to be ‘fixed-parameter tractable (FPT)’ if it has an algorithm that runs in time O(f(k)nc), where f is some (usually computable) function, and c is a constant that does not depend on k or n. What makes the theory more interesting is a hierarchy of intractable parameterized problem classes above FPT which helps in distinguishing those problems that are not fixed parameter tractable. Closely related to fixed-parameter tractability is the notion of preprocessing. A reduction to a problem kernel, or equivalently, problem kernelization means to apply a data reduction process in polynomial time to an instance (x,k) such that for the reduced instance (x,k) it holds that (x,k) is equivalent to (x,k), |x|g(k) and kg(k) for some function g only depending on k. Such a reduced instance is called a problem kernel. We refer to [5] for further details on parameterized complexity.

In this paper, we consider Unrestricted Fair FVS, Restricted Fair FVS, and Relaxed Fair FVS under structural parameters and solution size. Our results are as follows:

  • Unrestricted Fair FVS is W[1]-hard when parameterized by the treewidth or treedepth of the input graph.

  • Unrestricted Fair FVS admits an algorithm with running time O(3k),1 where k is the neighbourhood diversity of the input graph.

  • Restricted Fair FVS admits a kernel of size O(Δk). Here, Δ is the maximum degree of the graph, and k is the solution size.

  • Relaxed Fair FVS admits an algorithm with running time O(82k), where k is the solution size. This is our main technical result.

Related Work. FVS is extremely well studied in the realms of parameterized complexity. Downey and Fellows [6], and Bodlaender [3] proposed the first FPT algorithm with the running time O(17(k4)!). After a series of improvements, in 2010, Cao et al. gave an FPT algorithm running in time O(3.83k) [4]. The fastest known randomized algorithm is given by Li and Nederlof [14] and runs in time O(2.7k). Recently, Iwata and Kobayashi [7] announced the fastest known deterministic algorithm for FVS running in time O(3.460k).

The study of fair deletion problems in the realm of parameterized complexity was initiated by Masařík and Toufar [16] and Knop et al. [11]. Among several results, they showed that Unrestricted Fair Vertex Cover problem is W[1]-hard when parameterized by a combined parameter of the treedepth and the feedback vertex set number of the graph. Kolman et al. introduced MSO with fair objective functions [12]. It was further generalized by Masařík and Toufar [16] and later extended by Knop et al. [10]. Jacob et al. [8] studied Restricted d-Hitting Set and Restricted Fair FVS parameterized by solution size. For Unrestricted Fair FVS they designed an FPT algorithm parameterized by solution size and the treewidth of the input graph and using this they designed an FPT algorithm for Restricted Fair FVS running in time O(kO(k)).

Section snippets

Preliminaries

Throughout the paper, we follow the following notions. Let G be a graph, V(G) and E(G) denote the vertex set and the edge set of G, respectively. Let n and m denote the number of vertices and the number of edges of G, respectively. For a graph G and a set XV(G), by GX we denote the graph G induced on V(G)X. By NG(v), we denote the neighbourhood of v in G and by NG[v] we denote the closed neighbourhood of v in G. Degree of a vertex v in graph G is denoted by dG(v). The subscript in the

Unrestricted Fair Feedback Vertex Set

It is clear that if we do not put any restriction on the size of feedback vertex set S of a graph, then any graph has a trivial feedback vertex set S=V(G). However, we observe that, in Unrestricted Fair FVS problem, though we dropped the constraint on the size of S, the problem does not become easy. In this section, we present a parameterized reduction from Fair Vertex Cover (Fair VC) problem to the Unrestricted Fair FVS problem. In the Fair VC problem, we are given a graph G=(V,E) and a

Restricted Fair Feedback Vertex Set

In this section, we give a kernel for Restricted Fair FVS with respect to parameter Δ+k, where Δ is the maximum degree in the graph and k is the size of restricted fair feedback vertex set. We first give a couple of reductions to reduce an instance (G,k,) of Restricted Fair FVS problem.

Reduction Rule 1

If G contains a vertex v of degree at most 1, delete v. The new instance is (G{v},k,).

Reduction Rule 1 is safe, because for a given instance (G,k,) of Restricted Fair FVS, if the graph G has a vertex of

Relaxed Fair Feedback Vertex Set

In this section, we give an FPT algorithm for the Relaxed Fair FVS problem with respect to the solution size. An instance (G,k) of FVS can be trivially reduced to an instance (G,k,k) of Relaxed Fair FVS problem. It is easy to see that (G,k) is a yes-instance of FVS if and only if (G,k,k) is a yes-instance of Relaxed Fair FVS problem. This shows that Relaxed Fair FVS is NP-hard. We give an FPT algorithm for Relaxed Fair FVS using solution size as a parameter. Towards designing an algorithm for

Conclusion

In this work, we introduced three variants of the classical Feedback Vertex Set (FVS) problem: Unrestricted Fair FVS, Restricted Fair FVS, and Relaxed Fair FVS, and studied the problems with respect to structural parameters and the solution size. We proved that Unrestricted Fair FVS is W[1]-hard when parameterized by treewidth; Restricted Fair FVS is FPT when parameterized by the solution size and maximum degree and Relaxed Fair FVS is FPT when parameterized by the solution size. It would be

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

  • This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant no. 819416), and the Swarnajayanti Fellowship (Department of Science and Technology) grant DST/SJF/MSA-01/2017-18.

  • The second author's research was supported in part by the Science and Engineering Research Board (SERB), Govt. of India, under Sanction Order No. MTR/2018/001025.

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A preliminary version has been accepted for publication in the proceedings of the 15th International Computer Science Symposium in Russia (CSR), 2020.

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