SIAM Journal on Discrete Mathematics
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Edge domination on bipartite permutation graphs and cotriangulated graphs
Information Processing Letters
Approximation algorithms
A 2-approximation algorithm for the minimum weight edge dominating set problem
Discrete Applied Mathematics
A Unified Approach to Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs
ESA '94 Proceedings of the Second Annual European Symposium on Algorithms
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
A $(2 - c \frac{\log {n}}{n})$ Approximation Algorithm for the Minimum Maximal Matching Problem
Approximation and Online Algorithms
A better approximation ratio for the vertex cover problem
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Hi-index | 0.89 |
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-clog|V||V|)-approximation algorithm, where c is an arbitrary constant. In this paper, we present a (2-1@g^'(G))-approximation algorithm based on an LP relaxation, where @g^'(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@g^'(G))-approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2-1@D(G)+1, where @D(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@D(G). Moreover, @g^'(G) is @D(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.