Edge Dominating Set Problem Research Articles

Algorithms and hardness results for edge total domination problem in graphs

For a graph G=(V,E) without an isolated edge, a set D⊆E is an edge total dominating set of G if every edge e∈E is adjacent to at least one edge of D. The Minimum Edge Total Dominating Set problem is to compute a minimum cardinality edge total dominating set of G. It is known that the decision version of the problem is NP-complete for bipartite graphs, chordal graphs, and planar graphs. On the positive side, the problem is efficiently solvable for trees. In this paper, we further study the complexity of this problem in other graph classes. We design a linear-time algorithm to compute a minimum cardinality edge total dominating set in chain graphs, a subclass of bipartite graphs. We also propose linear-time algorithms for two subclasses of chordal graphs, namely, split graphs and proper interval graphs with no cut vertices. In addition, we show that the problem is APX-complete for graphs with maximum degree 3 and propose an approximation algorithm for the problem in k-regular graphs, where k≥4. We discuss the complexity difference between the Minimum Edge Dominating Set problem and Minimum Edge Total Dominating Set problem which seem to be closely related.

Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation and inapproximability results.

Minimum-Cost b-Edge Dominating Sets on Trees

In this paper, we consider the minimum-cost b-edge dominating set problem. This is a generalization of the edge dominating set problem, but the computational complexity for trees is an astonishing open problem. We make steps toward the resolution of this open problem in the following three directions. (1) We give the first combinatorial polynomial-time algorithm for paths. Prior to our work, the polynomial-time algorithm for paths used linear programming, and it was known that the linear-programming approach could not be extended to trees. Thus, our algorithm would yield an alternative approach to a possible polynomial-time algorithm for trees. (2) We give a fixed-parameter algorithm for trees with the number of leaves as a parameter. Thus, a possible NP-hardness proof for trees should make use of trees with unbounded number of leaves. (3) We give a fully polynomial-time approximation scheme for trees. Prior to our work, the best known approximation factor was two. If the problem is NP-hard, then a possible proof cannot be done via a gap-preserving reduction from any APX-hard problem unless P $$=$$ NP.

The computational complexity of dominating set problems for instances with bounded minors of constraint matrices

We consider boolean linear programming formulations of the vertex and edge dominating set problems and prove their polynomial-time solvability for classes of graphs with constraint matrices having bounded minors in the absolute value.

Maximum matching and kernelization of edge dominating set

The edge dominating set problem is NP-hard, even when the graph is restricted to planar or bipartite graphs with maximum degree three. In this paper, we prove that in every graph where each vertex is incident to at most one vertex of degree one, the cardinality of maximum matching is at least 2|V|/(3+max⁡(3,Δ(G))). Using the aforementioned result together with a simple reduction rule, we obtain a linear kernel of size 6k for the edge dominating set problem for graphs with maximum degree three.

The computational complexity of three graph problems for instances with bounded minors of constraint matrices

We consider boolean linear programming formulations of the independent set, the vertex and the edge dominating set problems and prove their polynomial-time solvability for classes of graphs with (augmented) constraint matrices having bounded minors in the absolute value.

On Approximating (Connected) 2-Edge Dominating Set by a Tree

The edge dominating set problem (EDS) is to compute a minimum size edge set such that every edge is dominated by some edge in it. This paper considers a variant of EDS with extensions of multiple and connected dominations combined. In the b-EDS problem, each edge needs to be dominated b times. Connected EDS requires an edge dominating set to be connected while it has to form a tree in Tree Cover. Although each of EDS, b-EDS, and Connected EDS (or Tree Cover) has been well studied, each known to be approximable within 2 (or 8/3 for b-EDS in general), nothing is known when these extensions are imposed simultaneously on EDS unlike in the case of the (vertex) dominating set problem. We consider Connected 2-EDS and 2-Tree Cover (i.e., a combination of 2-EDS and Tree Cover), and present a polynomial algorithm approximating each within 2. Moreover, it will be shown that the single tree computed is no larger than twice the optimum for (not necessarily connected) 2-EDS, thus also approximating 2-EDS equally well. It also implies that 2-EDS with clustering properties can be approximated within 2 as well.

Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a naive linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal–dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem. These results match the currently known best results for purely edge-weighted graphs.

An Improved Algorithm for Parameterized Edge Dominating Set Problem

An edge dominating set of a graph G = (V, E) is a subset M ⊆ E of edges such that each edge in E \M is incident to at least one edge in M. In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O*(2.2351k)-time and polynomial-space algorithm. This is an improvement over the previous best time bound of O*(2.3147k). We also show two corollaries: the parameterized weighted edge dominating set problem can be solved in O*(2.2351k) time and polynomial space; and a minimum edge dominating set of a graph G can be found in O*(1.7957τ) time where τ is the size of a minimum vertex cover of G.

A refined exact algorithm for Edge Dominating Set

In this paper, we present a new exact algorithm for the Edge Dominating Set problem, and analyze its running time by the Measure and Conquer method. Our algorithm runs in 1.3160nnO(1) time for a graph with n vertices, which is the currently fastest known time for the Edge Dominating Set problem. By designing new branching rules based upon conceptually simple local structures, which we call clique-producing vertices and cycles, we obtain an algorithm that is simpler than the previously fastest known algorithm and has an improved time bound as well.

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

An edge dominating set in a graph G = (V, E) is a subset S of edges such that each edge in E ? S is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NP-hard problem that has been extensively studied in approximation algorithms and parameterized complexity. In this paper, we present improved hardness results and parameterized approximation algorithms for edge dominating set. More precisely, we first show that it is NP-hard to approximate edge dominatingset in polynomial time within a factor better than 1.18. Next, we give a parameterized approximation schema (with respect to the standard parameter) for the problem and, finally, we develop an O ?(1.821 ? )-time exact algorithm where ? is the size of a minimum vertex cover of G.

Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dominating Set problems on graphs of bounded clique-width. We show that on an $n$-vertex graph of clique-width $t$ both problems (1) cannot be solved in time $f(t)n^{o(t)}$ for any function $f$ of $t$ unless exponential time hypothesis fails, and (2) can be solved in time $n^{O(t)}$.

Lower bounds for local approximation

In the study of deterministic distributed algorithms, it is commonly assumed that each node has a unique O (log n )-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO- checkable graph optimisation problems ; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem . By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4--1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs : We say that a graph is (α, r )-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius- r neighbourhoods. We show that there exists a finite (α, r )-homogeneous 2 k -regular graph of girth at least g for any α < 1 and any r , k , and g .

On Total Unimodularity of Edge–Edge Adjacency Matrices

We consider total unimodularity for edge–edge adjacency matrices that represent adjacency relations between pairs of edges in a graph. These matrices appear in integer programming formulations of the minimum maximal matching problem, the edge dominating set problem, and so on. We investigate the problem of characterizing graphs that have totally unimodular edge–edge adjacency matrices, and give a necessary and sufficient condition for characterization. This condition is the first characterization for total unimodularity of edge–edge adjacency matrices.

Exact Algorithms for Annotated Edge Dominating Set in Graphs with Degree Bounded by 3

Given a graph G = (V, E) together with a nonnegative integer requirement on vertices r : V → Z+, the annotated edge dominating set problem is to find a minimum set M ⊆ E such that, each edge in E − M is adjacent to some edge in M, and M contains at least r(v) edges incident on each vertex v ∈ V . The annotated edge dominating set problem is a natural extension of the classical edge dominating set problem, in which the requirement on vertices is zero. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the annotated edge dominating set problem in graphs with maximum degree 3 can be solved in O∗(1.2721n) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O∗(2.2306k)-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an annotated edge dominating set of size k or not. key words: edge dominating sets, exact algorithms, cubic graphs

Memetic Algorithm with Filtering Scheme for the Minimum Weighted Edge Dominating Set Problem

The minimum weighted edge dominating set problem (MWEDS) generalizes both the weighted vertex cover problem and the problem of covering the edges of graph by a minimum cost set of both vertices and edges. In this paper, we propose a meta heuristic approach based on genetic algorithm and local search to solve the MWEDS problem. Therefore, the proposed method is considered as a memetic search algorithm which is called Memetic Algorithm with filtering scheme for minimum weighted edge dominating set, and called shortly (MAFS). In the MAFS method, three new fitness functions are invoked to effectively measure the solution qualities. The search process in the proposed method uses intensification scheme, called “filtering”, beside the main genetic search operations in order to achieve faster performance. The experimental results proves that the proposed method is promising in solving the MWEDS problem. (Less)

New parameterized algorithms for the edge dominating set problem

An edge dominating set of a graph G=(V,E) is a subset M⊆E of edges in the graph such that each edge in E−M is incident with at least one edge in M. In an instance of the parameterized edge dominating set problem, we are given a graph G=(V,E) and an integer k, and we are asked to decide whether G has an edge dominating set of size at most k. In this paper, we show that the parameterized edge dominating set problem can be solved in O∗(2.3147k) time and polynomial space. We also show that this problem can be reduced to a quadratic kernel with O(k3) edges.

Maximum $r$-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds

We show that for a fixed $r$, the number of maximal $r$-regular induced subgraphs in any graph with $n$ vertices is upper bounded by $\mathcal{O}(c^n)$, where $c$ is a positive constant strictly less than $2$. This bound generalizes the well-known result of Moon and Moser, who showed an upper bound of $3^{n/3}$ on the number of maximal independent sets of a graph on $n$ vertices. We complement this upper bound result by obtaining an almost tight lower bound on the number of (possible) maximal $r$-regular induced subgraphs possible in a graph on $n$ vertices. Our upper bound results are algorithmic. That is, we can enumerate all the maximal $r$-regular induced subgraphs in time $\mathcal{O}(c^n n^{\mathcal{O}(1)})$. A related question is that of finding a maximum-sized $r$-regular induced subgraph. Given a graph $G=(V,E)$ on $n$ vertices, the Maximum $r$-Regular Induced Subgraph (M-$r$-RIS) problem asks for a maximum-sized subset of vertices, $R \subseteq V$, such that the induced subgraph on $R$ is $r$-regular. As a by-product of the enumeration algorithm, we get a $\mathcal{O}(c^n)$ time algorithm for this problem for any fixed constant $r$, where $c$ is a positive constant strictly less than $2$. Furthermore, we use the techniques and results obtained in the paper to obtain improved exact algorithms for a special case of the Induced Subgraph Isomorphism problem, namely, the Induced $r$-Regular Subgraph Isomorphism problem, where $r$ is a constant, the $\delta$-Separating Maximum Matching problem and the Efficient Edge Dominating Set problem.

Approximating edge dominating set in dense graphs

We study the approximation complexity of the Minimum Edge Dominating Set problem in everywhere ϵ -dense and average ϵ ̄ -dense graphs. More precisely, we consider the computational complexity of approximating a generalization of the Minimum Edge Dominating Set problem, the so called Minimum Subset Edge Dominating Set problem. As a direct result, we obtain for the special case of the Minimum Edge Dominating Set problem in everywhere ϵ -dense and average ϵ ̄ -dense graphs by using the techniques of Karpinski and Zelikovsky, the approximation ratios of min { 2 , 3 / ( 1 + 2 ϵ ) } and of min { 2 , 3 / ( 3 − 2 1 − ϵ ̄ ) } , respectively. On the other hand, we give new approximation lower bounds for the Minimum Edge Dominating Set problem in dense graphs. Assuming the Unique Game Conjecture, we show that it is NP-hard to approximate the Minimum Edge Dominating Set problem in everywhere ϵ -dense graphs with a ratio better than 2 / ( 1 + ϵ ) with ϵ > 1 / 3 and 2 / ( 2 − 1 − ϵ ̄ ) with ϵ ̄ > 5 / 9 in average ϵ ̄ -dense graphs.

Erratum to: Linear Time Algorithms for Generalized Edge Dominating Set Problems

Erratum to: Linear Time Algorithms for Generalized Edge Dominating Set Problems