A Refined Exact Algorithm for Edge Dominating Set

Xiao, Mingyu; Nagamochi, Hiroshi

doi:10.1007/978-3-642-29952-0_36

Cited by 11 publications

(8 citation statements)

References 18 publications

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“…For a cycle c 1 c 2 c 3 c 4 of length 4, we can also branch with (3) by including either {c 1 , c 3 } or {c 2 , c 4 } into V 1 . For the details about the proof of this fact, reader is referred to [21,25,23].…”

Section: An Improved Parameterized Approximation Schemamentioning

confidence: 99%

“…When the graph is restricted to be of maximum degree 3, the result can be further improved to O * (2.1479 k ) [24]. There is also a long list of contributions to exact algorithms for edge dominating set, such as the O * (1.4423 |V | )-time algorithm by Raman et al [20], the O * (1.4082 |V | )-time algorithm by Fomin et al [15], the O * (1.3226 |V | )-time algorithm by Rooij and Bodlaender [21], and finally the O * (1.3160 |V | )-time algorithm by Xiao and Nagamochi [25].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2014

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Abstract. 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 dominating set 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 τ )-time exact algorithm where τ is the size of a minimum vertex cover of G.

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Section: An Improved Parameterized Approximation Schemamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2014

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“…Using the treewidth of graphs, Fomin et al [3] obtained an O * (1.4082 n )-time and exponentialspace algorithm. Analyzing with the measure and conquer method, van Rooij and Bodlaender [10] designed an O * (1.3226 n )-time and polynomial-space algorithm and later Xiao and Nagamochi [14] presented an O * (1.3160 n )-time and polynomial-space algorithm, which currently attains the best time bound to Minimum Edge Dominating Set. For graphs of maximum degree 3, an O * (1.2721 n )-time and polynomial-space algorithm is designed by Xiao and Nagamochi [15].…”

Section: Introductionmentioning

confidence: 99%

An Exact Algorithm for Lowest Edge Dominating Set

Iwaide

Nagamochi

2017

IEICE Trans. Inf. & Syst.

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SUMMARYGiven an undirected graph G, an edge dominating set is a subset F of edges such that each edge not in F is adjacent to some edge in F, and computing the minimum size of an edge dominating set is known to be NP-hard. Since the size of any edge dominating set is at least half of the maximum size μ(G) of a matching in G, we study the problem of testing whether a given graph G has an edge dominating set of size μ(G)/2 or not. In this paper, we prove that the problem is NP-complete, whereas we design an O * (2.0801 μ(G)/2 )-time and polynomial-space algorithm to the problem.

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“…Fomin et al [7] claimed an O * (1.4082 n )-time algorithm by considering the treewidth of the graphs. Rooij and Bodlaender [19] got an O * (1.3226 n )-time algorithm by using the 'measure and conquer' method, which was further improved to O * (1.3160 n ) [24]. In terms of parameterized algorithms with parameter k being the size of the solution, there are also a long list of contributions to the upper bound of the running time.…”

Section: Introductionmentioning

confidence: 99%

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

Xiao

Nagamochi

2013

IEICE Trans. Inf. & Syst.

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SUMMARYGiven 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 * (1.2721 n ) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O * (2.2306 k )-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an annotated edge dominating set of size k or not.

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A Refined Exact Algorithm for Edge Dominating Set

Cited by 11 publications

References 18 publications

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

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

An Exact Algorithm for Lowest Edge Dominating Set

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

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