Connected Vertex Cover Research Articles

Some observations on holographic algorithms

We define the notion of diversity for families of finite functions and express the limitations of a simple class of holographic algorithms, called elementary algorithms, in terms of limitations on diversity. We show that this class of elementary algorithms is too weak to solve the Boolean circuit value problem, or Boolean satisfiability, or the permanent. The lower bound argument is a natural but apparently novel combination of counting and algebraic dependence arguments that is viable in the holographic framework. We go on to describe polynomial time holographic algorithms that go beyond the elementarity restriction in the two respects that they use exponential size fields, and multiple oracle calls in the form of polynomial interpolation. These new algorithms, which use bases of three components, compute the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case, the parity can also be computed for any one slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes.

Enumeration and maximum number of minimal connected vertex covers in graphs

Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson (1979). Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well-studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the number of minimal connected vertex covers of a graph is at most 1.8668n, and these sets can be enumerated in time O(1.8668n). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.

Dynamic Parameterized Problems

We study the parameterized complexity of various graph theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties. In real world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic $$\Pi $$ -Deletion problem which is the dynamic variant of the classical $$\Pi $$ -Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic $$\Pi $$ -Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with $$1.0822^k n^{\mathcal {O}(1)}$$ running time and Dynamic Feedback Vertex Set has a randomized algorithm with $$1.6667^k n^{\mathcal {O}(1)}$$ running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with $$2^k n^{\mathcal {O}(1)}$$ running time and show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

Boundary classes for graph problems involving non-local properties

We continue the study of boundary classes for NP-hard problems and focus on seven NP-hard graph problems involving non-local properties: Hamiltonian Cycle, Hamiltonian Cycle Through Specified Edge, Hamiltonian Path, Feedback Vertex Set, Connected Vertex Cover, Connected Dominating Set and GraphVCconDimension. Our main result is the determination of the first boundary class for Feedback Vertex Set. We also determine boundary classes for Hamiltonian Cycle Through Specified Edge and Hamiltonian Path and give some insights on the structure of some boundary classes for the remaining problems.

Complexity and algorithms for the connected vertex cover problem in 4-regular graphs

In the connected vertex cover (CVC) problem, we are given a connected graph G and required to find a vertex cover set C with minimum cardinality such that the induced subgraph G[C] is connected. In this paper, we restrict our attention to the CVC problem in 4-regular graphs. We proved that the CVC problem is still NP-hard for 4-regular graphs and gave a lower bound for the problem. Moreover, we proposed two approximation algorithms for CVC problem with approximation ratio 32 and 43+O(1n), respectively.

A PTAS for minimum weighted connected vertex cover $$P_3$$ P 3 problem in 3-dimensional wireless sensor networks

Given a connected and weighted graph $$G=(V, E)$$G=(V,E) with each vertex v having a nonnegative weight w(v), the minimum weighted connected vertex cover $$P_{3}$$P3 problem $$(MWCVCP_{3})$$(MWCVCP3) is required to find a subset C of vertices of the graph with minimum total weight, such that each path with length 2 has at least one vertex in C, and moreover, the induced subgraph G[C] is connected. This kind of problem has many applications concerning wireless sensor networks and ad hoc networks. When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. In this paper, we propose a new concept called weak c-local and give the first polynomial time approximation scheme (PTAS) for $$MWCVCP_{3}$$MWCVCP3 in unit ball graphs when the weight is smooth and weak c-local.

On the parameterized complexity of dynamic problems

In a dynamic version of a (base) problem X it is assumed that some solution to an instance of X is no longer feasible due to changes made to the original instance, and it is required that a new feasible solution be obtained from what “remained” from the original solution at a minimal cost. In the parameterized version of such a problem, the changes made to an instance are bounded by an edit-parameter, while the cost of reconstructing a solution is bounded by some increment-parameter.Capitalizing on the recent initial work of Downey et al. on the Dynamic Dominating Set problem, we launch a study of the dynamic versions of a number of problems including Vertex Cover, Maximum Clique, Connected Vertex Cover and Connected Dominating Set. In particular, we show that Dynamic Vertex Cover is W[1]-hard, and the connected versions of both Dynamic Vertex Cover and Dynamic Dominating Set become fixed-parameter tractable with respect to the edit-parameter while they remain W[2]-hard with respect to the increment-parameter. Moreover, we show that Dynamic Independent Dominating Set is W[2]-hard with respect to the edit-parameter.We introduce the reoptimization parameter, which bounds the difference between the cardinalities of initial and target solutions. We prove that, while Dynamic Maximum Clique is fixed-parameter tractable with respect to the edit-parameter, it becomes W[1]-hard if the increment-parameter is replaced with the reoptimization parameter.Finally, we establish that Dynamic Dominating Set becomes W[2]-hard when the target solution is required not to be larger than the initial one, even if the edit parameter is exactly one.

A PTAS for the minimum weight connected vertex cover [formula omitted] problem on unit disk graphs

Let G=(V,E) be a weighted graph, i.e., with a vertex weight function w:V→R+. We study the problem of determining a minimum weight connected subgraph of G that has at least one vertex in common with all paths of length two in G. It is known that this problem is NP-hard for general graphs. We first show that it remains NP-hard when restricted to unit disk graphs. Our main contribution is a polynomial time approximation scheme for this problem if we assume that the problem is c-local and the unit disk graphs have minimum degree of at least two.

A Completeness Theory for Polynomial (Turing) Kernelization

The framework of Bodlaender et al. (J Comput Sys Sci 75(8):423---434, 2009) and Fortnow and Santhanam (J Comput Sys Sci 77(1):91---106, 2011) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, some issues are not addressed by this framework, including the existence of Turing kernels such as the "kernelization" of leaf out-branching $$(k)$$ ( k ) that outputs $$n$$ n instances each of size poly $$(k)$$ ( k ) . Observing that Turing kernels are preserved by polynomial parametric transformations (PPTs), we define two kernelization hardness hierarchies by the PPT-closure of problems that seem fundamentally unlikely to admit efficient Turing kernelizations. This gives rise to the MK- and WK-hierarchies which are akin to the M- and W-hierarchies of parameterized complexity. We find that several previously considered problems are complete for the fundamental hardness class WK[1], including Min Ones $$d$$ d -SAT $$(k)$$ ( k ) , Binary NDTM Halting $$(k)$$ ( k ) , Connected Vertex Cover $$(k)$$ ( k ) , and Clique parameterized by $$k \log n$$ k log n . We conjecture that no WK[1]-hard problem admits a polynomial Turing kernel. Our hierarchy subsumes an earlier hierarchy of Harnik and Naor that, from a parameterized perspective, is restricted to classical problems parameterized by witness size. Our results provide the first natural complete problems for, e.g., their class $$VC_1$$ V C 1 ; this had been left open.

The Price of Connectivity for Vertex Cover

Graph Theory The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class Θ₂^P = P^NP[\log], the class of decision problems that can be solved in polynomial time, provided we can make O(\log n) queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.

A 9k kernel for nonseparating independent set in planar graphs

We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. We focus on the Planar Maximum Nonseparating Independent Set problem, where given a planar graph and an integer k one has to find an independent set of size k whose removal leaves the graph connected. Our main result is a kernel of size at most 9k vertices for this problem. A direct consequence of this result is that Planar Connected Vertex Cover has no kernel with at most (9/8−ϵ)k vertices, for any ϵ>0, assuming P≠NP.As a by-product we show extremal graph theory results which might be of independent interest. We prove that graphs that contain no separator consisting of only degree two vertices contain (a) a spanning tree with at least n/4 leaves and (b) a nonseparating independent set of size at least n/9 (also, equivalently, a connected vertex cover of size at most 89n). The result (a) is a generalization of a theorem of Kleitman and West [12] who showed the same bound for graphs of minimum degree three. Finally, we show that every n-vertex outerplanar graph contains an independent set I and a collection of vertex-disjoint cycles C such that 9|I|⩾4n−3|C|.

Towards optimal kernel for connected vertex cover in planar graphs

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP⊆coNP/poly), for planar graphs Guo and Niedermeier [ICALP’08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS’11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this question in the negative: we show a 113k-vertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in the search for an optimal kernel.In our analysis, we show an extension of a theorem of Nishizeki and Baybars [Takao Nishizeki, Ilker Baybars, Lower bounds on the cardinality of the maximum matchings of planar graphs, Discrete Mathematics 28 (3) (1979) 255–267] which might be of independent interest: every planar graph with n≥3 vertices of degree at least 3 contains a matching of cardinality at least n≥3/3.

Planar graph vertex partition for linear problem kernels

A simple partition of the vertex set of a graph is introduced to analyze kernels for planar graph problems in which vertices and edges not in a solution have small distance to the solution. This method directly leads to improved kernel sizes for several problems, without needing new reduction rules. Moreover, new kernelization algorithms are developed for Connected Vertex Cover, Edge Dominating Set, and Maximum Triangle Packing problems, further improving the kernel sizes for these problems.

Kernelization hardness of connectivity problems in [formula omitted]-degenerate graphs

A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs.Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d≥2 unless NP⊆coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics–Colourful Graph Motif–analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via a reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.We extend our analysis by showing that, unless NP⊆coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d≥2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique Ki,j as a subgraph. As a d-degenerate graph cannot contain Kd+1,d+1 as a subgraph, the results holds also for graphs of bounded degeneracy.

PTAS for the minimum k-path connected vertex cover problem in unit disk graphs

In the Minimum k-Path Connected Vertex Cover Problem (MkPCVCP), we are given a connected graph G and an integer k ? 2, and are required to find a subset C of vertices with minimum cardinality such that each path with length k ? 1 has a vertex in C, and moreover, the induced subgraph G[C] is connected. MkPCVCP is a generalization of the minimum connected vertex cover problem and has applications in many areas such as security communications in wireless sensor networks. MkPCVCP is proved to be NP-complete. In this paper, we give the first polynomial time approximation scheme (PTAS) for MkPCVCP in unit disk graphs, for every fixed k ? 2.

Parameterized Measure & Conquer for Problems with No Small Kernels

Measure & Conquer (M&C) is a prominent technique for analyzing exact algorithms for computationally hard problems, in particular, graph problems. It tries to balance worse and better situations within the algorithm analysis. This has led, e.g., to algorithms for Minimum Vertex Cover with a running time of $\mathcal{O}(c^{n})$ for some constant c≈1.2, where n is the number of vertices in the graph. Several obstacles prevent the application of this technique in parameterized algorithmics, making it rarely applied in this area. However, these difficulties can be handled in some situations. We will exemplify this with two problems related to Vertex Cover, namely Connected Vertex Cover and Edge Dominating Set. For both problems, several parameterized algorithms have been published, all based on the idea of first enumerating minimal vertex covers. Using M&C in this context will allow us to improve on the hitherto published running times. In contrast to some of the earlier suggested algorithms, ours will use polynomial space.

Approximating Subdense Instances of Covering Problems

Abstract We study the approximability of subdense instances of various covering optimization problems, including Vertex Cover, Connected Vertex Cover, Set Cover, and Steiner Tree problems. In those instances, the minimum (or average) degree of the underlying graph is o ( n ) , but Ω ( n / ψ ( n ) ) for some function ψ of the number n of vertices. We design new approximation algorithms or new polynomial time approximation schemes (PTAS) for those problems and establish the first approximation hardness results for some of them.

PTAS for minimum weighted connected vertex cover problem with c-local condition in unit disk graphs

Given a graph G =( V , E ) with node weight w : V R +, the minimum weighted connected vertex cover problem (MWCVC) is to seek a subset of vertices of the graph with minimum total weight, such that ...

Connected vertex covers in dense graphs

We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worst-case ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2 / ( 1 + ε ) in graphs with average degree ε n , and give a family of near-tight examples.

Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.