Fully Dynamic (1+ e)-Approximate Matchings

We present the first data structures that maintain near optimal maximum cardinality and maximum weighted matchings on sparse graphs in sub linear time per update. Our main result is a data structure that maintains a (1+ε) approximation of maximum matching under edge insertions/deletions in worst case Õ(? mε-2) time per update. This improves the 3/2 approximation given by Neiman and Solomon [20] which runs in similar time. The result is based on two ideas. The first is to re-run a static algorithm after a chosen number of updates to ensure approximation guarantees. The second is to judiciously trim the graph to a smaller equivalent one whenever possible. We also study extensions of our approach to the weighted setting, and combine it with known frameworks to obtain arbitrary approximation ratios. For a constant ε and for graphs with edge weights between 1 and N, we design an algorithm that maintains an (1+ε) approximate maximum weighted matching in Õ(? m log N) time per update. The only previous result for maintaining weighted matchings on dynamic graphs has an approximation ratio of 4. 9108, and was shown by An and et al. [2], [3].

Authors

• Manoj Gupta 0002
• Richard Peng

Keywords

• Approximation algorithms
• Algorithm design and analysis
• Heuristic algorithms
• Approximation methods
• Data structures
• Optimization
• Computer science
• Data Structure
• Edge Weights
• Maximum Weight
• Approximate Ratio
• Matching Time
• Maximum Matching
• Sparse Graph
• Dynamic Graph
• Number Of Updates
• Time Graph
• Static Algorithm
• Weight Matching
• Running Time
• Undirected
• Estimation Algorithm
• Update Step
• Update Time
• Size Matching
• Entire Graph
• Graph Matching
• Approximate Matching
• Vertex Cover
• Single Update
• Static Graph
• Unweighted Graph
• dynamic algorithms
• matching
• approximation