Results on the min-sum vertex cover problem

Journal of Discrete Algorithms, 2009

We consider the concepts of a t-total vertex cover and a t-total edge cover (t ≥ 1), which generalise the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present N P-completeness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem. * A preliminary version of this 2 Preliminary observations involving α 0,t (G) and α 1,t (G)

2016

Given a graph G(V,E) of order n and a constant k 6 n, the max k-vertex cover problem consists of determining k vertices that cover the maximum number of edges in G. In its (standard) parameterized version, max k-vertex cover can be stated as follows: “given G, k and parameter l, does G contain k vertices that cover at least l edges?”. We first devise moderately exponential exact algorithms for max k-vertex cover, with complexity exponential to n (note that the known results concerned time bounds of the form n) by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, interestingly enough, although max k-vertex cover is non fixed parameter tractable with respect to l, it is fixed parameter tractable with respect to the size τ of a minimum vertex cover of G. We also point out that the same happens for a lot of well-known problems quite different from max k-vertex cover. We finally study approximation of max k-vertex cover by moderately ...

Information Processing Letters, 1989

2010

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψ k (G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining ψ k (G) is NP-hard for each k ≥ 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of ψ k (G) and provide several estimations and exact values of ψ k (G). We also prove that ψ 3 (G) ≤ (2n + m)/6, for every graph G with n vertices and m edges.

Information Sciences Letters

The minimum vertex cover (VC) problem is to find a minimum number of vertices in an undirected graph such that every edge in the graph is incident to at least one of these vertices. This problem is a classical NP-hard combinatorial optimization problem with applications in a wide range of areas. Hence, there is a need to develop approximate algorithms to find a small VC in a reasonable time. This paper presents a new construction algorithm for the minimum VC problem. Extensive experiments on benchmark graphs show that the proposed algorithm is extremely competitive and complementary to existing construction algorithms for minimum VC problem.

2002

2005

Tables 1 through 4 summarize the experiments for the VCP. The first two columns in these tables refer to the number of vertices and the number of edges in the input graph. The column Optimum has the optimum value and the cpu time to solve the instance. Tables 1 and 3 show results for the Greedy algorithm when this algorithm was executed 10 times. Recall that this algorithm is not deterministic.

Congressus …, 2007

We define two related graph covering problems motivated by net-work monitoring. Both problems relate to the idea of graph mea-surement from a single vertex via shortest paths, and are meant to serve as lower and upper bounds on the amount of information that can be ...

Journal of the Indonesian Mathematical Society, 2016

In this paper we have computed minimum covering Seidel energies of a star graph, complete graph, crown graph, complete bipartite graph and cocktail party graphs. Upper and lower bounds for minimum covering Seidel energies of graphs are also established.

Operations Research Letters, 2006

We provide a new LP relaxation of the maximum vertex cover problem and a polynomial-time algorithm that finds a solution within the approximation factor 1 − 1/(2q), whereq is the size of the smallest clique in a given clique-partition of the edge weighting of G.

Genetic Algorithms within the Framework of …, 1994

2016

Evolution of large scale networks demand for efficient way of communication in the networks. One way to propagate information in the network is to find vertex cover. In this paper we describe a variant of vertex cover problem naming it N-distance Vertex Minimal Cover(N-MVC) Problem to optimize information propagation throughout the network. A minimum subset of vertices of a unweighted and undirected graph G = (V, E) is called N-MVC if for all v in V , v is at distance less than or equal to N from at least one of the the vertices in N-MVC. In the following paper, this problem is defined, formulated and an approximation algorithm is proposed with discussion on its correctness and upper bound.

Discrete Applied Mathematics, 2011

Motivated by applications in software programming, we consider the problem of covering a graph by a feasible labeling. Given an undirected graph G = (V , E), two positive integers k and t, and an alphabet Σ, a feasible labeling is defined as an assignment of a set L v ⊆ Σ to each vertex v ∈ V , such that (i) |L v | ≤ k for all v ∈ V and (ii) each label α ∈ Σ is used no more than t times. An edge e = {i, j} is said to be covered by a feasible labeling if L i ∩ L j ̸ = ∅. G is said to be covered if there exists a feasible labeling that covers each edge e ∈ E. In general, we show that the problem of deciding whether or not a tree can be covered is strongly NP-complete. For k = 2, t = 3, we characterize the trees that can be covered and provide a linear time algorithm for solving the decision problem. For fixed t, we present a strongly polynomial algorithm that solves the decision problem; if a tree can be covered, then a corresponding feasible labeling can be obtained in time polynomial in k and the size of the tree. For general graphs, we give a strongly polynomial algorithm to resolve the covering problem for k = 2, t = 3.

Journal of Combinatorial Theory, Series A, 1998

For every xed graph H , we determine the H -covering number of K n , for all n > n 0 (H). We prove that if h is the number of edges of H , and gcd(H ) = d is the greatest common divisor of the degrees of H , then there exists n 0 = n 0 (H), such that for all n > n 0 , C (H; K n ) = d dn 2h d n ? 1 d ee;

Algorithmic Operations Research, 2011

In this paper, we consider the classical NP-complete VERTEX COVER problem in large graphs. We assume that the size and the access to the input graph impose the following constraints: (1) the input graph must not be modified (integrity of the input instance), (2) the computer running the algorithm has a memory of limited size (compared to the graph) and