Elsevier

Theoretical Computer Science

Volume 860, 8 March 2021, Pages 98-116
Theoretical Computer Science

Paths to trees and cacti,☆☆

https://doi.org/10.1016/j.tcs.2021.01.033Get rights and content

Abstract

We know that Tree Contraction does not admit a polynomial kernel unless NPcoNP/poly, while Path Contraction admits a kernel with O(k) vertices. The starting point of this article is the following natural questions: What is the structure of the family of paths that allows Path Contraction to admit a polynomial kernel? Apart from the size of the solution, what other additional parameters should we consider so we can design polynomial kernels for these basic contraction problems? To design polynomial kernels, we consider the family of trees with the bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study Bounded Tree Contraction. Here, an input is a graph G, integers k and , and the goal is to decide whether, there is a subset FE(G) of size at most k such that G/F is a tree with at most leaves. We design a kernel with O(k) vertices and O(k2+k) edges for this problem. We complement this result by giving kernelization lower bound. We also prove similar results for Bounded Out-Tree Contraction and Bounded Cactus Contraction.

Introduction

Graph editing problems are one of the central problems in graph theory that have received a lot of attention in the realm of parameterized complexity. Some important graph editing operations are vertex deletion, edge deletion, edge addition, and edge contraction. For a family of graphs F, the F-Editing problem takes as an input a graph G and an integer k, and the aim is to decide if at most k edit operations can result in a graph that belongs to the graph family F. In fact, the F-Editing problem, where the edit operations are restricted to vertex deletion or edge deletion or edge addition or edge contraction alone has also been studied extensively in parameterized complexity. When we just focus on deletion operation (vertex/edge deletion) then the corresponding problem is called F-Vertex (Edge) Deletion problem. For instance, the F-Editing problem encompasses several NP-hard problems such as Vertex Cover, Feedback vertex set, Planar F-Deletion, Interval Vertex Deletion, Chordal Vertex Deletion, Odd cycle transversal, Edge Bipartization, Tree Contraction, Path Contraction, Split Contraction, Clique Contraction etc. However, most of the study in parametrized complexity or classical complexity, has been restricted to combination of vertex deletion, edge deletion or edge addition [9], [7], [8], [6], [19], [21], [18], [26], [28], [30], [33], [14], [15], [16], [22], [2], [3], [35]. Also, see the recent survey by Crespelle et al. [10]. Only recently, edge contraction as an edit operation has gained attention in the realm of parameterized complexity. In this paper we study three edge-contraction problems from the perspective of kernelization complexity – one of the established sub-area in parameterized complexity.

For several families of graphs F, early papers by Watanabe et al. [36], [37] and Asano and Hirata [1] showed that F-Edge Contraction is NP-complete. In the framework of parameterized complexity (or even the classical complexity), these problems exhibit properties that differ greatly from those of problems where we only delete or add vertices and edges. For instance, deleting k edges from a graph such that the resulting graph is a tree is polynomial-time solvable. On the other hand, Asano and Hirata showed that Tree Contraction is NP-hard [1]. A well-known result by Cai [4] states that in a case F is a hereditary family of graphs with a finite set of forbidden induced subgraphs, then the graph modification problem defined by F and the edit operations restricted to vertex deletion, edge deletion, and edge addition admits a simple FPT algorithm. Indeed, for these problems, the result by Cai [4] does not hold when the edit operation is edge contraction. In particular, Lokshtanov et al. [32] and Cai and Guo [5] independently showed that if F is either the family of P-free graphs for some 5 or the family of C-free graphs for some 4, then F-Edge Contraction is W[2]-hard. To the best of our knowledge, Heggernes et al. [25] were the first to explicitly study F-Edge Contraction from the viewpoint of Parameterized Complexity. They showed that in case F is the family of trees, F-Edge Contraction is FPT but does not admit a polynomial kernel, while in case F is the family of paths, the corresponding problem admits a faster algorithm and an O(k)-vertex kernel. Golovach et al. [23] proved that if F is the family of planar graphs, then F-Edge Contraction is again FPT. Moreover, Cai and Guo [5] showed that in case F is the family of cliques, F-Edge Contraction is solvable in time 2O(klogk)nO(1), while in case F is the family of chordal graphs, the problem is W[2]-hard. Heggernes et al. [27] developed an FPT algorithm for the case where F is the family of bipartite graphs. Later, a faster algorithm was proposed by Guillemot and Marx [24].

It is clear from our discussion that the complexity of the graph editing problem when restricted to edge contraction seems to be more difficult than their vertex or edge deletion counterparts. The starting point of our research is the following result by Heggernes et al. [25] who showed that Tree Contraction does not admit a polynomial kernel unless NPcoNP/poly [25] and Path Contraction admits a linear vertex kernel.

We wanted to understand the structure of the family of paths that allows Path Contraction to admit a polynomial kernel. Apart from the size of the solution, what other additional parameters should we consider so we can design polynomial kernels for these basic contraction problems? One of the natural candidates for such an extension is to consider the family of trees with the bounded number of leaves. With the goal to apprehend the understanding of the role the number of leaves plays in the kernelization complexity for contracting to the “path-like” graph, we study the problem which we call as Bounded Tree Contraction (Bounded TC). Formally, we define the problem as follows.

We give a kernel for Bounded TC with O(k) vertices and O(k2+k) edges. The approach we follow is similar to the one Heggernes et al. [25] used to obtain a linear kernel for Path Contraction. We observe that our algorithm works even when the input is a directed graph. In particular, we consider Bounded Out-Tree Contraction (Bounded OTC), which is defined as follows.

By incorporating direction appropriately into our algorithm for Bounded TC, we get a kernel for Bounded OTC with O(k2+k) vertices and arcs.

We also study the contraction problem for a class of graphs which generalizes trees – the family of cactus. Formally, the problem we study is defined as follows.

The number of leaves of cactus is defined as the number of leaves in its block decomposition. For Bounded CC we give a kernel with O(k2+k) vertices and edges.

Finally, we complement all our kernelization algorithms by giving lower bounds. In particular, we show that Bounded TC, Bounded OTC and Bounded CC do not admit a kernel of size O((k2+k)1ϵ) kernels unless NPcoNP/poly for any ϵ>0.

Section snippets

Preliminaries

Graph theory  We consider graphs with a finite number of vertices. For an undirected graph G, by V(G) and E(G) we denote the set of vertices and edges of G respectively. For a directed graph (or digraph) D, by V(D) and A(D) we denote the sets of vertices and directed edges (arcs) in D, respectively. Two vertices u,v are said to be adjacent in G (or in D) if there is an edge (arc) uvE(G) (or in A(D)) and u,v are said to be endpoints of the edge (arc) uv. The neighborhood of a vertex v, denoted

Kernel for Bounded Tree Contraction

In this section we design a kernelization algorithm for Bounded TC. We first present some preliminary results. For every integer 2, consider a set of trees which have at most leaves. The following observation states that this set of graphs is closed under edge contraction.

Observation 2

Consider a tree T with leaves. Let T be the graph obtained from T contracting an edge in T. Then, T is a tree with at most leaves.

This set is also closed under an operation of uncontracting an edge with some

Kernel for Bounded Out-Tree Contraction

In this section, we design a kernelization algorithm for Bounded OTC. We start with some preliminary results regarding out-tree. Digraph obtained by subdividing an arc of out-tree results in another out-tree. The operation of subdividing an arc uv in D is comprises a deletion of the arc uv and addition of a new vertex w as an out-neighbor of u and an in-neighbor of v.

Observation 3

Consider an out-tree T with leaves. Let T be the digraph obtained from T by one of the following operations.

  • 1.

    subdividing an arc;

Kernel for Bounded Cactus Contraction

In this section, we design a kernelization algorithm for Bounded Cactus Contraction. We start with some known properties of cactus.

Observation 4

The following statements hold for a cactus T.

  • 1.

    |E(T)|2|V(T)|

  • 2.

    Every vertex of degree at least 3 is a cut-vertex.

Proof

For a cactus T, let D be its block decomposition.

(1) We prove this using the induction on the number of blocks in a cactus graph. Our induction hypothesis is: if the number of blocks in T is strictly less than q then |E(T)|2|V(T)|. For the base case,

Kernel lower bounds

In this section we present a lower bound for the kernels presented in Sections 3, 4, and 5 under the assumption that NPcoNP/poly. The problem Dominating Set takes as an input a graph and an integer k, and the goal is to decide whether the input graph contains a dominating set of size at most k. We can encode any instance with O(n2) bits where n is the number of vertices in the input graph. Jansen and Pieterse proved that Dominating Set does not admit a compression of bit-size O(n2ϵ), for any ϵ

Conclusion

In this article, we analyze the structure of the family of paths that allows Path Contraction to admit a polynomial kernel but forbids Tree Contraction. Apart from solution size k, we make the number of leaves, , as an additional parameter to bridge the gap between kernels of these two problems. We call this problem as Bounded Tree Contraction. We present a polynomial kernel for this problem. We also prove that this kernel is optimal under a certain complexity assumption. We prove similar

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

The authors would like to thank Prof. Daniel Lokshtanov for helpful discussions.

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  • Cited by (3)

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    Preliminary version of this article has appeared in 10th International Conference on Algorithms and Complexity (CIAC 2017).

    ☆☆

    The research leading to these results has received funding from the European Research Council (ERC) via grant PARAPPROX, reference 306992. The first and the third authors are supported by the PBC Program of Fellowships for Outstanding Post-doctoral Researchers from China and India (No. 5101479000); and Horizon 2020 Framework Programme, ERC Consolidator Grant LOPPRE (No. 819416), respectively.

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