Parameterized algorithms for finding highly connected solution☆
Introduction
Vertex deletion (subset) problems form an important sub-area of graph optimization problems. An input to a prototype vertex deletion problem consists of a graph G and an integer ℓ and the objective is to find a vertex subset S of size at most ℓ such that satisfies a property, such as being an edgeless graph (Vertex Cover), an acyclic graph (FVS), a bipartite graph (OCT), a chordal graph (CVD), a planar graph (PVD), and a (topological) minor-free graph. In literature, several variants of these classical vertex deletion problems are considered. The most notable ones include those where we demand that is connected, λ-edge-connected (that is, for every pair of vertices in S there are at least λ edge-disjoint paths in ) and an edgeless graph (S is an independent set). A classical result by Lewis and Yannakakis [7] shows that most of the vertex deletion problems are NP-complete and so are its variants [12], [13]. These problems have been studied extensively from the perspective of the Approximation Algorithms and the Parameterized Complexity to overcome these intractability results.
We first take a detour and give the basic definitions from Parameterized Complexity. The goal of parameterized complexity is to find ways of solving NP-hard problems more efficiently than brute force: here the aim is to restrict the combinatorial explosion to a parameter that is hopefully much smaller than the input size. Formally, a parameterization of a problem is assigning an integer k to each input instance, and we say that a parameterized problem is fixed-parameter tractable (FPT) if there is an algorithm that solves the problem in time , where is the size of the input and f is an arbitrary computable function depending on the parameter k only. For more background, the reader is referred to the monographs [1], [3], [5], [9].
Problem and Parameterization. To introduce our question and the parameterization, we fix a concrete vertex subset problem, namely, the classical Vertex Cover problem. In this vertex subset problem, the graph is an edgeless graph. In other words, the set S of size at most ℓ must include at least one end-point of every edge of G. When we demand that is connected or more general λ-edge-connected then the problem is called Connected Vertex Cover (CVC) or more generally λ-Edge-Connected VC (λ-ECVC), respectively. While, the study of CVC is quite old in Parameterized Complexity [6], only recently Einarson et al. [4] studied λ-ECVC and designed lossy kernel as well as an algorithm with running time . In some sense, the algorithm for λ-ECVC is the starting point of our work and one of our main motivations. A question that triggered this work was the following:
Let us call this version of Vertex Cover as HC-VC. Observe that when we are seeking (n-k)-edge-connected subgraph, then the size of S is at least , as every vertex in must have at least neighbors. So, if we apply the algorithm of Einarson et al. [4], then we get an algorithm with running time . On the other hand, since S contains all but at most vertices of , there is an algorithm running in time , that tries all vertex subsets of size at least as a potential solution. Given an algorithm with running time , a natural question that arises is the following.
Also, dense connected subgraph problems have been studied from social networking point of view [8] [11].
The above algorithm for HC-VC, that runs in time, does not use any property of vertex cover! It seamlessly works for HC-FVS, HC-OCT, HC-CVD and HC-PVD. In fact, this algorithm also works for domination (Dominating Set) as well as cut (Multiway Cut, Multicut) problems. In Dominating Set (DS), we seek S such that every vertex in has a neighbor in S. In Multiway Cut, apart from G and an integer ℓ, we are given a vertex subset , called terminals, and the objective is to find an ℓ-sized vertex subset S, such that in there is no path from s to t, for any pair of vertices . In Multicut, apart from G and an integer ℓ, we are given t pairs of terminals , and the objective is to find a ℓ-sized vertex subset S, such that in there is no path from to , . Thus, naturally we ask whether HC-FVS, HC-OCT, HC-CVD, HC-PVD, HC-DS, HC-Multiway Cut and HC-Multicut are FPT.
Our Results and Methods. We show that HC-VC, HC-FVS, HC-OCT, HC-PVD, HC-DS, and HC-Multiway Cut are FPT. To design some of our FPT algorithms we consider a generic vertex deletion problem, whose specific instantiation leads to HC-VC, HC-FVS, and HC-PVD. Let be a family of graphs. In the -Deletion problem, we need to ensure that does not contain any graph in as a minor (a graph L is a minor of , if it can be obtained from by vertex deletions, edge deletions and edge contractions). If is an edge, or a triangle, or a and , then it corresponds to HC-VC, HC-FVS, and HC-PVD, respectively. The main idea of the algorithms for -Deletion problems is as follows. Let H be the subset of vertices in G such that the degree of every vertex in H is at least . We find a constant size subset of vertices of H, say Z, that does not belong to S, but whose all but number of common neighbors do belong to S. Since the size of common neighbors of Z is at least , we have that all but vertices get fixed. For the remaining vertices, we can guess which one of them belongs to S in time, leading to the desired FPT algorithm.
For HC-DS and HC-Multiway Cut we need additional ideas. We first show that a graph G with n vertices and given integer k and , is (n-k)-edge-connected if and only if for every vertex . This helps us in characterizing the solution S as a subset where every vertex has degree at least . Furthermore, we need an algorithm for HC-Steiner Subgraph, as a subroutine. Here, given a graph G, positive integers ℓ, k and a subset of terminals T, the objective is to find a vertex subset S of size at most ℓ such that is (n-k)-edge-connected and . We show that HC-Steiner Subgraph admits an algorithm with running time . Using this as a subroutine we show that HC-DS and HC-Multiway Cut admit time algorithms. We also prove that HC-Multicut problem is W[1]-hard.
Section snippets
Preliminaries
We first set up notations and give a characterization of (n-k)-edge-connected subgraph.
Notations. Let G be a graph. We use and to denote the set of vertices and edges of G, respectively. Throughout the paper we use n and m to denote and , respectively. For a set S, by , we mean . For a set of vertices , denote . For a vertex v, we use to denote the set of its neighbors, and use to denote . We use to denote the smallest
Vertex subset problems
In this section, we give two simple algorithms for HC-VC and HC-FVS, that illustrate the idea of “common neighbors branching”. Then we provide an algorithm for HC--Deletion.
Steiner subgraph
In this section, we present an FPT algorithm for HC-Steiner Subgraph. An algorithm for this problem is used as a subroutine for an algorithm for HC-Multicut and HC-DS. The problem itself is defined as follows.
In what follows, we prepare ourselves to give an FPT algorithm for HC-Steiner Subgraph. Recall and . From Theorem 1 we know, if there exists a solution S, . This leads to the following simple reduction rule.
Reduction Rule 1 Let be an instance of HC-Steiner
Dominating set and multiway cut
In this section we give algorithms for (n-k)-edge-connected version of the classical Dominating Set and Multiway Cut problems.
Vertex multicut
In this section, we give a hardness reduction for (n-k)-edge-connected version of the classical problem. Let us first define the problem formally.
Here we give a reduction from the problem to HC-Multicut.
Theorem 9 The HC-Multicut is -.
Proof Let be an problem instance, where . We assume that the parameter is for , for simplicity. The question is whether there exists a subset of vertices of cardinality (at least) such that any two vertices in
Conclusion
In this paper, we designed FPT algorithms for the highly connected versions of several natural graph problems, with the parameter being the distance from being “ connected”. Developing polynomial kernels or showing the nonexistence of polynomial kernels remains an interesting direction to pursue.
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.
Acknowledgements
We thank anonymous referees of an earlier version of the paper for several suggestions. Especially for finding a fatal flaw and giving suggestions for improving the running time of the algorithm.
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This article belongs to Section A: Algorithms, automata, complexity and games, Edited by Paul Spirakis.