Exact Algorithms for Edge Domination

Abstract

An edge dominating set in a graph G = (V, E) is a subset of the edges D ⊆ E such that every edge in E is adjacent or equal to some edge in D. The problem of finding an edge dominating set of minimum cardinality is NP-hard. We present a faster exact exponential time algorithm for this problem. Our algorithm uses O(1.3226 n ) time and polynomial space. The algorithm combines an enumeration approach of minimal vertex covers in the input graph with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwise. In each of these refinement steps, the worst cases in the measure and conquer analysis of the current algorithm are reconsidered and a new branching strategy is proposed on one of these worst cases. In this way a series of algorithms appears, each one slightly faster than the previous, ending in the O(1.3226 n ) time algorithm. For each algorithm in the series, we also give a lower bound on its running time.We also show that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226 n ) time and polynomial space using modifications of the algorithm for edge dominating set. In addition we consider the matrix dominating set problem which we solve in O(1.3226 n+m ) time and polynomial space, and the parametrised minimum weight maximal matching problem for which we obtain an O * (2.4178 k ) time algorithm.

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

“…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].…”

“…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].…”

Inexact graph matching using a hierarchy of matching processes

“…The high computational complexity of current optimal methods poses a significant problem, which we attempt to address in this paper. Indeed, there has been some recent interest in reducing the computational complexity of inherently NP-hard problems, particularly in the field of graph theory [16][17][18][19][20].…”

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

“…Research on exponential-time algorithms for some natural and basic problems, such as independent set [9], [20], coloring [1], exact satisfiability [3] and so on, has a long history. Recently, some other basic graph problems, such as dominating set [8], edge dominating set [19] and feedback set [17], also have drawn much attention in this line of research. Furthermore, to get more understanding of the structural properties of NP-complete problems, people also have interests in exactly solving problems in sparse and low-degree graphs.…”

“…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.…”