Elsevier

Theoretical Computer Science

Volume 942, 9 January 2023, Pages 47-56
Theoretical Computer Science

Parameterized algorithms for finding highly connected solution

https://doi.org/10.1016/j.tcs.2022.11.024Get rights and content

Abstract

To introduce our question and the parameterization, consider the classical Vertex Cover problem. In this problem, the input is a graph G on n vertices and a positive integer , and the goal is to find a vertex subset S of size at most such that GS is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be nk edge-connected. Clearly, the problem is NP-complete, as substituting k=n1, we obtain the Connected Vertex Cover problem. A simple observation also shows that the problem admits an algorithm with running time nO(k). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most , such that G[S] is nk edge-connected and GS satisfies a property. We show that this version of well-known problems such as Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal and Multiway Cut admit an algorithm with running time f(k)nO(1), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for nk edge connected Steiner Subgraph, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for Multicut.

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 and the objective is to find a vertex subset S of size at most such that GS 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 G[S] is connected, λ-edge-connected (that is, for every pair of vertices in S there are at least λ edge-disjoint paths in G[S]) 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 f(k)|I|O(1), where |I| 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 GS is an edgeless graph. In other words, the set S of size at most must include at least one end-point of every edge of G. When we demand that G[S] 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 2O(λ)nO(1). 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 nk+1, as every vertex in G[S] must have at least nk neighbors. So, if we apply the algorithm of Einarson et al. [4], then we get an algorithm with running time 2O((nk)2)nO(1). On the other hand, since S contains all but at most k1 vertices of V(G), there is an algorithm running in time nO(k), that tries all vertex subsets of size at least nk+1 as a potential solution. Given an algorithm with running time nO(k), 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 nO(k) 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 GS has a neighbor in S. In Multiway Cut, apart from G and an integer , we are given a vertex subset TV(G), called terminals, and the objective is to find an -sized vertex subset S, such that in GS there is no path from s to t, for any pair of vertices s,tT. In Multicut, apart from G and an integer , we are given t pairs of terminals (si,ti), and the objective is to find a -sized vertex subset S, such that in GS there is no path from si to ti, i{1,,t}. 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 F be a family of graphs. In the F-Deletion problem, we need to ensure that GS does not contain any graph in LF as a minor (a graph L is a minor of GS, if it can be obtained from GS by vertex deletions, edge deletions and edge contractions). If F is an edge, or a triangle, or a K5 and K3,3, then it corresponds to HC-VC, HC-FVS, and HC-PVD, respectively. The main idea of the algorithms for F-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 nk. We find a constant size subset of vertices of H, say Z, that does not belong to S, but whose all but O(1) number of common neighbors do belong to S. Since the size of common neighbors of Z is at least n|Z|k, we have that all but O(k) vertices get fixed. For the remaining O(k) vertices, we can guess which one of them belongs to S in 2O(k) 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 n>2k, is (n-k)-edge-connected if and only if for every vertex vV(G),deg(v)nk. This helps us in characterizing the solution S as a subset where every vertex has degree at least nk. Furthermore, we need an algorithm for HC-Steiner Subgraph, as a subroutine. Here, given a graph G, positive integers , k and a subset of terminals T, the objective is to find a vertex subset S of size at most such that G[S] is (n-k)-edge-connected and TS. We show that HC-Steiner Subgraph admits an algorithm with running time 2O(klogk)nO(1). Using this as a subroutine we show that HC-DS and HC-Multiway Cut admit 2O(klogk)nO(1) 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 V(G) and E(G) to denote the set of vertices and edges of G, respectively. Throughout the paper we use n and m to denote |V(G)| and |E(G)|, respectively. For a set S, by GS, we mean G[V(G)S]. For a set of vertices AV(G), denote A=V(G)A. For a vertex v, we use N(v) to denote the set of its neighbors, and use deg(v) to denote |N(v)|. We use δ(G) 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-F-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 VL={u|deg(u)<nk} and VH={u|deg(u)nk}. From Theorem 1 we know, if there exists a solution S, VLS=. This leads to the following simple reduction rule.

Reduction Rule 1

Let (G,T,k,) 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 W[1]-hard.

Proof

Let I=(G,k1) be an

problem instance, where |V(G)|=n. We assume that the parameter is k1 for
, for simplicity. The question is whether there exists a subset of vertices of cardinality (at least) k1 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 “n1 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.

References (13)

There are more references available in the full text version of this article.

This article belongs to Section A: Algorithms, automata, complexity and games, Edited by Paul Spirakis.

View full text