Edge Dominating Set Problem Research Articles

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP -hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G = ( V , E ) , the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a ( 2 − c log | V | | V | ) -approximation algorithm, where c is an arbitrary constant. In this paper, we present a ( 2 − 1 χ ′ ( G ) ) -approximation algorithm based on an LP relaxation, where χ ′ ( G ) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of | V | . Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also ( 2 − 1 χ ′ ( G ) ) -approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2 − 1 Δ ( G ) + 1 , where Δ ( G ) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least 2 − 1 Δ ( G ) . Moreover, χ ′ ( G ) is Δ ( G ) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.

Approximation algorithms for partially covering with edges

The edge dominating set (EDS) and edge-cover (EC) problems are classical graph covering problems in which one seeks a minimum cost collection of edges which covers the edges or vertices, respectively, of a graph. We consider the generalized partial cover version of these problems, in which failing to cover an edge, in the EDS case, or vertex, in the EC case, induces a penalty. Given a bound on the total amount of penalties that we are permitted to pay, the objective is to find a minimum cost cover with respect to this bound. We give an 8/3-approximation for generalized partial EDS. This result matches the best-known guarantee for the { 0 , 1 } -EDS problem, a specialization in which only a specified set of edges need to be covered. Moreover, 8/3 corresponds to the integrality gap of the natural formulation of the { 0 , 1 } -EDS problem. Our techniques can also be used to derive an approximation scheme for the generalized partial edge-cover problem, which is NP -complete even though the uniform penalty version of the partial edge-cover problem is in P .

Linear Time Algorithms for Generalized Edge Dominating Set Problems

We prove that a generalization of the edge dominating set problem, in which each edge e needs to be covered b e times for all e∈E, admits a linear time 2-approximation for general unweighted graphs and that it can be solved optimally for weighted trees. We show how to solve it optimally in linear time for unweighted trees and for weighted trees in which b e ∈{0,1} for all e∈E. Moreover, we show that it solves generalizations of weighted matching, vertex cover, and edge cover in trees.

Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

We give substantially improved exact exponential-time algorithms for a number of NP-hard problems. These algorithms are obtained using a variety of techniques. These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique. These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented. We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time. The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems. The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results. For example, for the Minimum Maximal Matching we give an O*(1.4425n) algorithm improving the previous best result of O*(1.4422m) [35]. For the Odd Cycle Transversal problem, we give an O*(1.62n) algorithm which improves the previous time bound of O*(1.7724n) [3].

Approximability of the capacitated [formula omitted]-edge dominating set problem

In this paper, we discuss the approximability of the capacitated b -edge dominating set problem, which generalizes the edge dominating set problem by introducing capacities and demands on the edges. We present an approximation algorithm for this problem and show that it achieves a factor of 8/3 for general graphs and a factor of 2 for bipartite graphs. Moreover, we discuss the relationships of the edge dominating set problem and the vertex cover problem. The results show that improving the approximation factor beyond 8/3 using our approach of adding valid inequalities to a natural linear programming relaxation is as hard as improving the approximation factor for vertex cover beyond 2.

A 2-approximation algorithm for the minimum weight edge dominating set problem

We present a polynomial-time algorithm approximating the minimum weight edge dominating set problem within a factor of 2. It has been known that the problem is NP-hard but, when edge weights are uniform (so that the smaller the better), it can be efficiently approximated within a factor of 2. When general weights were allowed, however, very little had been known about its approximability, and only very recently was it shown to be approximable within a factor of 2 1 10 by reducing to the edge cover problem via LP relaxation. In this paper we extend the approach given therein, by studying more carefully polyhedral structures of the problem, and obtain an improved approximation bound as a result. While the problem considered is as hard to approximate as the weighted vertex cover problem is, the best approximation (constant) factor known for vertex cover is 2 even for the unweighted case, and has not been improved in a long time, indicating that improving our result would be quite difficult.

On approximability of the independent/connected edge dominating set problems

We investigate polynomial-time approximability of the problems related to edge dominating sets of graphs. When edges are unit-weighted, the edge dominating set problem is polynomially equivalent to the minimum maximal matching problem, in either exact or approximate computation, and the former problem was recently found to be approximable within a factor of 2 even with arbitrary weights. It will be shown, in contrast with this, that the minimum weight maximal matching problem cannot be approximated within any polynomially computable factor unless P= NP . The connected edge dominating set problem and the connected vertex cover problem also have the same approximability when edges/vertices are unit-weighted. The former problem is already known to be approximable, even with general edge weights, within a factor of 3.55. We will show that, when general weights are allowed, (1) the connected edge dominating set problem can be approximated within a factor of 3+ε, and (2) the connected vertex cover problem is approximable within a factor of lnn+3 but cannot be within (1−ε) lnn for any ε>0 unless NP⊂ DTIME(n O( log logn) ) .

Edge Dominating Sets in Graphs

We prove that the edge dominating set problem for graphs is $NP$-complete even when restricted to planar or bipartite graphs of maximum degree 3. We show as a corollary that the minimum maximal matching and the achromatic number problems are $NP$-complete. A new linear time algorithm for finding minimum independent edge dominating sets in trees is described, based on an observed relationship between edge dominating sets and independent sets in total graphs.